A viticulturist is investigating the survival of a new hybrid vine graft. The probability that a graft is successful is denoted by ppp. A sample of 100 grafts is taken, and the random variable S S\,S represents the number of successful grafts, where S∼B(100,p)S \sim \text{B}(100, p)S∼B(100,p).
Using a normal approximation, the probability that S S\,S is at least 53 is 0.0054 to 4 decimal places.
Show that p p\,p satisfies 426.01p2−446.01p+110.25=0426.01p^2 - 446.01p + 110.25 = 0426.01p2−446.01p+110.25=0 when normal probability tables are used.
Hence find the value of ppp, justifying your answer.