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inclusive."},{"text":"-1 and 1","correct":false,"explanation":"The scale is sometimes confused with scales that include negative values, but probabilities cannot be negative. Their minimum is 0 and their maximum is 1."},{"text":"0 and 100","correct":false,"explanation":"This mixes decimal probabilities with percentage probabilities. Percentages range from 0% to 100%, but probabilities in decimal form range from 0 to 1."},{"text":"1 and 10","correct":false,"explanation":"Probability is not measured on a ten-point scale. An impossible event has probability 0, while a certain event has probability 1."}]},{"id":"a37d064d-cab7-9bf5-80a1-bc5d1cdf99e6","cardType":"cloze","points":null,"front":"In probability, a value of 0 means the event is [ ] and a value of 1 means it is certain.","back":"In probability, a value of 0 means the event is impossible and a value of 1 means it is certain.","mcqOptions":[{"text":"impossible","correct":true,"explanation":"A probability of 0 means there is no chance that the event will occur, so the event is described as impossible."},{"text":"unlikely","correct":false,"explanation":"An unlikely event can still happen and therefore has a probability greater than 0. A probability of exactly 0 means the event is impossible."},{"text":"likely","correct":false,"explanation":"A likely event has a relatively high probability, not a probability of 0. A value of 0 specifically describes an impossible event."},{"text":"certain","correct":false,"explanation":"This reverses the ends of the probability scale. A certain event has probability 1, while an impossible event has probability 0."}]},{"id":"52e62ba2-2f2c-8e8e-9ed8-9fbe7ef4a910","cardType":"cloze","points":null,"front":"In probability, a value of 0 means the event is impossible and a value of 1 means it is [ ].","back":"In probability, a value of 0 means the event is impossible and a value of 1 means it is certain.","mcqOptions":[{"text":"certain","correct":true,"explanation":"A probability of 1 represents the maximum possible probability, meaning the event is guaranteed to happen and is therefore certain."},{"text":"likely","correct":false,"explanation":"A likely event has a high probability but may still fail to happen. A probability of exactly 1 means the event is certain."},{"text":"impossible","correct":false,"explanation":"This reverses the probability scale: an impossible event has probability 0. An event with probability 1 is certain to occur."},{"text":"unlikely","correct":false,"explanation":"An unlikely event has a low probability close to 0, not the maximum probability. A value of 1 means the event is certain."}]},{"id":"fef32fbd-06d6-4d1e-7501-7d0f4a57771e","cardType":"basic","points":2,"front":"What is the difference between an outcome and an event?","back":"

Outcome: one possible result (e.g. landing on blue).
Event: the specific result you are interested in (e.g. 'landing on blue').

","mcqOptions":[{"text":"

Outcome: one possible result (e.g. landing on blue).
Event: the specific result you are interested in (e.g. 'landing on blue').

","correct":true,"explanation":"An outcome is an individual possible result of an experiment. An event describes the result or collection of outcomes whose probability is being considered."},{"text":"

Outcome: the result you are interested in (e.g. landing on blue).
Event: every possible result of the experiment (e.g. 'all spinner colours').

","correct":false,"explanation":"This confuses an event with the full sample space. An outcome is one possible result, while an event is the result or outcomes being investigated."},{"text":"

Outcome: a collection of possible results (e.g. landing on a cool colour).
Event: one go of the experiment (e.g. 'spinning once').

","correct":false,"explanation":"One go of an experiment is a trial, not an event. An outcome is one possible result, while an event identifies the result or outcomes of interest."},{"text":"

Outcome: the chance of a result (e.g. a probability of 0.5).
Event: the equipment being used (e.g. 'a spinner').

","correct":false,"explanation":"A probability measures chance and the spinner is the experimental equipment. An outcome is a possible result, while an event is the result or outcomes being considered."}]},{"id":"45cd7c85-4390-a67e-e7ae-81a6d6608660","cardType":"basic","points":1,"front":"What does it mean if a dice or spinner is biased?","back":"The outcomes are not equally likely.","mcqOptions":[{"text":"The outcomes are not equally likely.","correct":true,"explanation":"A biased dice or spinner favours some results over others, so its possible outcomes do not all have the same probability."},{"text":"The outcomes are always equally likely.","correct":false,"explanation":"Equally likely outcomes describe a fair dice or spinner, which is the opposite of biased. With biased equipment, some outcomes are more likely than others."},{"text":"The outcomes are completely impossible.","correct":false,"explanation":"Bias changes the relative chances of outcomes; it does not necessarily make every outcome impossible. The defining feature is that the outcomes are not equally likely."},{"text":"The outcomes are known in advance.","correct":false,"explanation":"Bias does not make the next outcome predictable with certainty. It means only that the possible outcomes do not all have equal probabilities."}]},{"id":"ce20f5a3-80e3-f5e9-b08f-19218e6b0ade","cardType":"basic","points":1,"front":"If a table lists every possible outcome of an experiment, what must the probabilities add up to?","back":"1","mcqOptions":[{"text":"1","correct":true,"explanation":"Exactly one of the possible outcomes must occur, so the probabilities across a complete table of mutually exclusive outcomes must add up to 1."},{"text":"100","correct":false,"explanation":"This confuses decimal probabilities with percentages. Probabilities written as percentages total 100%, but probabilities written in the usual decimal or fractional form total 1."},{"text":"0","correct":false,"explanation":"A total of 0 would mean every possible outcome was impossible, so the experiment could not produce a result. All possible outcomes have probabilities totalling 1."},{"text":"2","correct":false,"explanation":"Adding to more than 1 would represent more than a 100% total chance. A complete set of possible outcomes must have probabilities that sum to 1."}]},{"id":"2382c3b4-72b9-3b00-77c2-22faf04af11c","cardType":"basic","points":2,"front":"How do you calculate a missing probability from a table of all possible outcomes?","back":"

Add all known probabilities.
Subtract that total from 1.

","mcqOptions":[{"text":"

Add all known probabilities.
Subtract that total from 1.

","correct":true,"explanation":"All probabilities in a complete outcome table total 1. Adding the known values and subtracting their sum from 1 therefore gives the missing probability."},{"text":"

Add all known probabilities.
Subtract that total from 100.

","correct":false,"explanation":"Subtracting from 100 incorrectly treats decimal probabilities as whole percentages. The known probabilities should be added, then their total should be subtracted from 1."},{"text":"

Multiply all known probabilities.
Subtract that product from 1.

","correct":false,"explanation":"Multiplication is used for some combined-event calculations, not for completing a probability table. Add the known probabilities and subtract their total from 1."},{"text":"

Add all known probabilities.
Use that total as the missing value.

","correct":false,"explanation":"The sum of the known probabilities is the probability already accounted for, not the missing value. Subtract that sum from 1 to find the remainder."}]},{"id":"72c70429-a760-08d6-14c3-e6d87b7d3eab","cardType":"cloze","points":null,"front":"A single go of an experiment (like one roll of a dice) is called a [ ].","back":"A single go of an experiment (like one roll of a dice) is called a trial.","mcqOptions":[{"text":"trial","correct":true,"explanation":"A trial is one performance or go of a probability experiment, such as rolling a dice once or spinning a spinner once."},{"text":"experiment","correct":false,"explanation":"An experiment is the overall process that can be performed, such as rolling a dice repeatedly. One individual go of that experiment is called a trial."},{"text":"outcome","correct":false,"explanation":"An outcome is the result produced, such as rolling a four, rather than the act of rolling. A single go of an experiment is a trial."},{"text":"event","correct":false,"explanation":"An event is the result or group of outcomes being investigated, not one performance of the experiment. A single go is called a trial."}]},{"id":"8d12aa89-a7ac-2084-e7bf-2639a5aefa4a","cardType":"basic","points":1,"front":"How do you calculate the estimated number of times an event will happen?","back":"$$$\\text{probability} \\times \\text{number of trials}$$","mcqOptions":[{"text":"$$$\\text{probability} \\times \\text{number of trials}$$","correct":true,"explanation":"Multiplying the probability by the number of trials gives the expected frequency, which estimates how many times the event should occur across all the trials."},{"text":"$$$\\text{probability} + \\text{number of trials}$$","correct":false,"explanation":"Adding quantities with different meanings does not produce an expected frequency. Multiply the probability by the number of trials to estimate how often the event will happen."},{"text":"$$$\\frac{\\text{number of trials}}{\\text{probability}}$$","correct":false,"explanation":"Dividing the number of trials by the probability makes less likely events appear more frequent. The correct calculation is probability multiplied by the number of trials."},{"text":"$$$\\frac{\\text{probability}}{\\text{number of trials}}$$","correct":false,"explanation":"Dividing by the number of trials produces a very small value rather than an expected frequency. Multiply the probability by the number of trials instead."}]},{"id":"860fcecb-2754-9a57-7e90-75e40c99d21b","cardType":"cloze","points":null,"front":"An estimate is a [ ], not a guarantee.","back":"An estimate is a sensible prediction, not a guarantee.","mcqOptions":[{"text":"sensible prediction","correct":true,"explanation":"An estimate uses available information to make a reasonable prediction, but the actual result may differ because probability describes uncertainty rather than certainty."},{"text":"certain result","correct":false,"explanation":"Treating an estimate as certain ignores the natural variation in random events. An estimate is a sensible prediction, so the actual result is not guaranteed."},{"text":"random guess","correct":false,"explanation":"A random guess need not use any evidence or probability. An estimate is a sensible prediction based on the information available, although it may not be exact."},{"text":"exact calculation","correct":false,"explanation":"An exact calculation would give a fixed result, whereas an estimate predicts an uncertain outcome. The missing term is sensible prediction because actual frequencies can vary."}]},{"id":"93bed78a-a6f1-5397-d82a-7b0908e0535c","cardType":"cloze","points":null,"front":"An estimate is a sensible prediction, not a [ ].","back":"An estimate is a sensible prediction, not a guarantee.","mcqOptions":[{"text":"guarantee","correct":true,"explanation":"An estimate predicts what is likely or expected, but random variation means the observed result can differ. It therefore cannot guarantee the actual outcome."},{"text":"prediction","correct":false,"explanation":"An estimate is itself a prediction based on available information, so it is wrong to exclude prediction. The sentence contrasts a sensible prediction with a guarantee."},{"text":"forecast","correct":false,"explanation":"A forecast is another type of prediction, so an estimate may reasonably be described as one. The intended contrast is that an estimate is not a guarantee."},{"text":"approximation","correct":false,"explanation":"An estimate can be an approximation rather than an exact value, so excluding approximation reverses its meaning. The missing word is guarantee because estimates remain uncertain."}]},{"id":"fe596584-077d-02d6-2fca-51bc881be6e0","cardType":"basic","points":1,"front":"What does mutually exclusive mean?","back":"Outcomes that cannot happen at the same time.","mcqOptions":[{"text":"Outcomes that cannot happen at the same time.","correct":true,"explanation":"Mutually exclusive outcomes have no overlap, so one trial cannot produce both outcomes. For example, a single coin toss cannot be both heads and tails."},{"text":"Outcomes that can happen at the same time.","correct":false,"explanation":"This describes overlapping outcomes, not mutually exclusive ones. Mutually exclusive outcomes cannot occur together in the same trial, so their intersection has probability zero."},{"text":"Outcomes that are equally likely to happen.","correct":false,"explanation":"Equal likelihood concerns whether outcomes have the same probability, not whether they overlap. Mutually exclusive outcomes are those that cannot happen at the same time."},{"text":"Outcomes that have probabilities adding to one.","correct":false,"explanation":"Probabilities add to one only when outcomes are both mutually exclusive and exhaustive. Mutual exclusivity alone means the outcomes cannot happen at the same time."}]},{"id":"64e54233-fbbe-64cb-4c54-72603237a1f7","cardType":"basic","points":1,"front":"How do you find the probability of one or another mutually exclusive outcome happening?","back":"Add their individual probabilities together.","mcqOptions":[{"text":"Add their individual probabilities together.","correct":true,"explanation":"For mutually exclusive outcomes, there is no overlap to count twice. Therefore, the probability of one or the other occurring is the sum of their probabilities."},{"text":"Multiply their individual probabilities together.","correct":false,"explanation":"Multiplication is commonly used for independent events linked by “and”, not mutually exclusive outcomes linked by “or”. Here, add the individual probabilities together."},{"text":"Subtract their individual probabilities together.","correct":false,"explanation":"Subtraction may be used for complements or differences, but it does not calculate either mutually exclusive outcome occurring. Their individual probabilities must be added together."},{"text":"Divide their individual probabilities together.","correct":false,"explanation":"Division compares probabilities rather than combining possible outcomes. Because the outcomes are mutually exclusive and connected by “or”, add their individual probabilities together."}]},{"id":"bddcfb35-6d2b-88e6-ca0b-cf6fc7519a33","cardType":"cloze","points":null,"front":"If outcomes are in the ratio $5:4:3$, the total number of parts is [ ].","back":"If outcomes are in the ratio $5:4:3$, the total number of parts is 12.","mcqOptions":[{"text":"12","correct":true,"explanation":"The total number of ratio parts is found by addition: $5+4+3=12$. Each outcome’s probability can then be written over this total."},{"text":"9","correct":false,"explanation":"This results from adding only $5+4$ and overlooking the third part. All ratio parts must be included, giving $5+4+3=12$."},{"text":"60","correct":false,"explanation":"Multiplying $5\\times4\\times3$ does not find the total number of ratio parts. Ratio parts are added, so the correct total is $12$."},{"text":"15","correct":false,"explanation":"This comes from multiplying the largest part, $5$, by the three outcomes. The parts can differ, so they must be added to obtain $12$."}]},{"id":"187807b2-f72b-0a4d-47f2-7e01d95dc1af","cardType":"basic","points":2,"front":"How do you convert a ratio into probabilities?","back":"

Add the ratio parts to find the total.
Put each part over the total as a fraction.

","mcqOptions":[{"text":"

Add the ratio parts to find the total.
Put each part over the total as a fraction.

","correct":true,"explanation":"The total number of ratio parts represents all possible outcomes. Dividing each individual part by that total gives its probability, and the resulting probabilities sum to one."},{"text":"

Multiply the ratio parts to find the product.
Put each part over the product as a fraction.

","correct":false,"explanation":"Multiplying ratio parts does not give the total share and produces probabilities that do not sum to one. Add the parts, then place each part over that total."},{"text":"

Add the ratio parts to find the total.
Put the total over each part as a fraction.

","correct":false,"explanation":"Reversing each fraction can produce values greater than one, which cannot be probabilities. After adding the parts, place each individual part over the total."},{"text":"

Count the ratio parts to find the outcomes.
Put each part over the outcomes as a fraction.

","correct":false,"explanation":"Using the number of categories as the denominator wrongly assumes equal probabilities. Add the numerical ratio parts, then divide each part by that total."}]},{"id":"61d34adc-195e-54a6-5861-0eb59528f442","cardType":"basic","points":1,"front":"If the probability of outcome A is twice the probability of outcome B, what ratio describes their relationship?","back":"$$2:1$","mcqOptions":[{"text":"$$2:1$","correct":true,"explanation":"If A is twice as probable as B, A has two equal probability parts for every one part belonging to B. Therefore, $A:B=2:1$."},{"text":"$$1:2$","correct":false,"explanation":"This reverses the order and makes B twice as probable as A. Since A has twice B’s probability, the correct ratio is $A:B=2:1$."},{"text":"$$2:2$","correct":false,"explanation":"This ratio simplifies to $1:1$, meaning A and B are equally probable. A must have two parts for every one part of B, giving $2:1$."},{"text":"$$1:1$","correct":false,"explanation":"A ratio of $1:1$ represents equal probabilities, not one probability being twice the other. Twice as likely gives the relationship $A:B=2:1$."}]},{"id":"9380f874-1e88-c368-5315-2a964468ec48","cardType":"cloze","points":null,"front":"If one missing probability is \"three times\" another, the leftover probability must be split into [ ] equal parts.","back":"If one missing probability is \"three times\" another, the leftover probability must be split into 4 equal parts.","mcqOptions":[{"text":"4","correct":true,"explanation":"The probabilities are in the ratio 3:1, so the ratio contains 3 + 1 = 4 equal parts altogether."},{"text":"3","correct":false,"explanation":"This counts only the three parts belonging to the larger probability. The smaller probability contributes one additional part, so the leftover must be divided into 4 equal parts."},{"text":"2","correct":false,"explanation":"Splitting into two equal parts would make the missing probabilities the same. A 3:1 relationship has four parts altogether, so the leftover must be divided into 4 parts."},{"text":"6","correct":false,"explanation":"Doubling the multiplier does not give the total number of ratio parts. The ratio is 3:1, whose entries sum to 4, so there are 4 equal parts."}]},{"id":"dfabadd2-2c6c-dba2-7fc5-16502ae4bba7","cardType":"cloze","points":null,"front":"When two missing probabilities are described as \"the same\", they are split in a [ ] ratio.","back":"When two missing probabilities are described as \"the same\", they are split in a 1:1 ratio.","mcqOptions":[{"text":"1:1","correct":true,"explanation":"Equal probabilities contain the same number of parts. Each receives one identical share, so their simplest ratio is 1:1."},{"text":"1:2","correct":false,"explanation":"This makes the second probability twice the first rather than making them equal. Equal probabilities each receive one identical share, giving the ratio 1:1."},{"text":"2:1","correct":false,"explanation":"This makes the first probability twice the second, reversing an unequal 1:2 relationship. The word “same” requires equal shares, represented by 1:1."},{"text":"0:1","correct":false,"explanation":"A zero part would make the first probability impossible rather than equal to the second. Two equal missing probabilities are split in the ratio 1:1."}]},{"id":"3c6f14ba-bf48-d9ac-51e8-e41372c9c95d","cardType":"basic","points":1,"front":"How do you find the total number of items if you know the count of one item and its probability?","back":"$$$\\frac{\\text{number of items}}{\\text{probability}}$$","mcqOptions":[{"text":"$$$\\frac{\\text{number of items}}{\\text{probability}}$$","correct":true,"explanation":"Probability equals the number of the chosen items divided by the total. Rearranging this relationship gives the total as the known number of items divided by its probability."},{"text":"$$$\\text{number of items} \\times \\text{probability}$$","correct":false,"explanation":"Multiplication finds an expected count when the total is already known. To recover the total from a known count and probability, divide the number of items by the probability."},{"text":"$$$\\frac{\\text{probability}}{\\text{number of items}}$$","correct":false,"explanation":"This reverses the required division and does not produce a total number of items. Rearranging probability = count ÷ total gives total = count ÷ probability."},{"text":"$$$\\frac{\\text{number of items}}{1-\\text{probability}}$$","correct":false,"explanation":"Using the complementary probability would estimate the total from the count of items not in the stated category. For the given category, divide its count by its probability."}]},{"id":"49e29c8f-243b-32f0-f63c-31a3cd4cb7d0","cardType":"cloze","points":null,"front":"If 30 green pens represent a probability of 0.15, the total number of pens is [ ].","back":"If 30 green pens represent a probability of 0.15, the total number of pens is 200.","mcqOptions":[{"text":"200","correct":true,"explanation":"The total is found by dividing the observed count by its probability: $30 \\div 0.15 = 200$. Therefore, the full collection contains 200 pens."},{"text":"4.5","correct":false,"explanation":"This results from multiplying $30$ by $0.15$, which would find 15% of 30 rather than the total. The required calculation is $30 \\div 0.15 = 200$."},{"text":"20","correct":false,"explanation":"This comes from treating $0.15$ as $1.5$, a decimal-place error. Dividing correctly by $0.15$ gives $30 \\div 0.15 = 200$ pens."},{"text":"2000","correct":false,"explanation":"This places the decimal one position too far when dividing by $0.15$. Since 30 is 15% of the total, the correct total is $30 \\div 0.15 = 200$."}]},{"id":"939a46a1-5de8-a003-1894-b2e88861cdf4","cardType":"basic","points":1,"front":"What is the probability of outcome $X$ written in mathematical notation?","back":"$$P(X)$","mcqOptions":[{"text":"$$P(X)$","correct":true,"explanation":"The notation $P(X)$ means the probability that outcome or event $X$ occurs, with the event written inside parentheses after $P$."},{"text":"$$P(X')$","correct":false,"explanation":"A prime commonly denotes the complement, so this represents the probability that $X$ does not occur. The probability of $X$ itself is written $P(X)$."},{"text":"$$n(X)$","correct":false,"explanation":"The notation $n(X)$ represents the number of elements or outcomes in set $X$, not its probability. The probability of the outcome is written $P(X)$."},{"text":"$$P(X^c)$","correct":false,"explanation":"A superscript $c$ denotes the complement of an event, so this is the probability that $X$ does not occur. The required notation is $P(X)$."}]},{"id":"5ea9f791-7204-6c21-c99a-ebbf72df2772","cardType":"basic","points":1,"front":"A gardener plants 50 seeds with a growth probability of 0.88. What is the estimated number of seeds that grow?","back":"44 ($0.88 \\times 50$)","mcqOptions":[{"text":"44 ($0.88 \\times 50$)","correct":true,"explanation":"Multiplying the number of trials by the growth probability gives the expected frequency: $50 \\times 0.88 = 44$, so an estimated 44 seeds grow."},{"text":"6 ($0.12 \\times 50$)","correct":false,"explanation":"This uses the failure probability, $1 - 0.88 = 0.12$, so it estimates seeds that do not grow. The growing estimate is $0.88 \\times 50 = 44$."},{"text":"45 ($0.9 \\times 50$)","correct":false,"explanation":"Rounding $0.88$ to $0.9$ before calculating loses accuracy and overestimates growth. Using the given probability directly gives $0.88 \\times 50 = 44$."},{"text":"43 ($0.88 \\times 49$)","correct":false,"explanation":"Subtracting one from the number of trials creates an off-by-one error. All 50 seeds are planted, so the estimate is $0.88 \\times 50 = 44$."}]}]}],"examCountdown":null,"heading":"Probability","title":"Probability Flashcards","topbar":{"backHref":"/gcse/maths/ccea/flashcards"},"topicSidebar":"$L39"}]}],"topicId":"6b09693d-17b1-20a2-02a8-3818d090bb60","walled":"$L3a"}],"$L3b","$L3c"]