A coin is tossed 431 times. Use Binomial Distribution to approximate the probability of getting at most 235 tails. How do I solve this without a calculator?

A coin is tossed 431 times. Use Binomial Distribution to approximate the probability of getting at most 235 tails. How do I solve this without a calculator?

Let X = Number of tails out of 431

p = proportion of getting tails = 1/2 = 0.5.

So X follows binomial distribution with parameters:

n = 431, p = 0.50.

Let's use normal approximation to the binomial.

Mean = µ = n*p = 431*0.5 = 215.5, σ = ?n∗p∗(1−p)?=431∗0.5∗0.5?? = 10.38.

a) Here we want to find the probability of getting at most 235 tails

That is P(X ≤ 235)

Using continuity correction, and then find the z-score.

?z=σx+0.5−μ?=10.38235+0.5−215.5?=1.926732203=1.93?

Using z-table, P(Z < 1.93) = 0.9732

So P(X ≤ 235) = P(Z < 1.93) = 0.9732

Answer: P(X ≤ 235) = 0.9732.