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2.3 Probability

2.3 Probability

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Question 143

The continuous random variable X X\,X represents the time, in minutes, a patient spends in a waiting room. The probability density function of X X\,X is given by

f(x)={k(36−x2)0≤x≤60otherwise f(x) = \begin{cases} k(36 - x^2) & 0 \le x \le 6 \\ 0 & \text{otherwise} \end{cases} f(x)={k(36−x2)0​0≤x≤6otherwise​

where k k\,k is a constant.

a.

Show that k=1144\displaystyle k = \frac{1}{144}k=1441​.

[2]
b.

Find the cumulative distribution function F(x)F(x)F(x) for 0≤x≤60 \le x \le 60≤x≤6.

[2]
c.

Find the probability that a patient waits for longer than 2 minutes.

[2]
d.

A patient has already been waiting for 2 minutes.

Find the probability that this patient will wait for at least 4 minutes in total.

[3]
e.

Four patients are selected at random.

Find the probability that exactly 3 of them waited for longer than 2 minutes.

[2]
Markscheme

2.3 Probability Questions

  1. A Level
  2. /Maths
  3. /2.3 Probability

200 exam-style questions on OCR (MEI) A Level Maths 2.3 Probability, covering 2.3.1 Calculate the probability of an event, 2.3.2 Complementary events, 2.3.3 Expected frequency of an event, 2.3.4 Diagrams to calculate probabilities, 2.3.5 Mutually exclusive and independent events, 2.3.6 Add probabilities for mutually exclusive events, 2.3.7 Multiply probabilities for independent events, 2.3.8 Mutually exclusive and independent events (notation) (A-level only), 2.3.9 Venn diagrams for probabilities (A-level only), 2.3.10 Conditional probabilities (A-level only), and 2.3.11 Independence via conditional probability (A-level only). Each one has a worked solution and a mark scheme showing where the marks go.

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