A fair coin is tossed until 100 heads appear. Find the probability that at least 226 tosses will be necessary.

Solution:

Given coin is tossed 100 times, n = 100

Since n = 100, the sample is a large sample.

We may use the normal distribution as an approximation to the binomial distribution to obtain the answer.

Let X denote the number of heads with parameters.

The probability of getting a head when a coin is tossed is p = 1/2

The mean of the binomial distribution is = np = 100(1/2) = 50

Standard deviation = √npq =√100(1/2)(1/2)

= √(100/4)

= √25

σ = 5

To find the probability that at least 226 tosses will be necessary. P(x ≥ 226)= ?

But by using normal distribution,

z = (x - μ)/σ

Where x = 226, μ = 50 and σ = 5 and n =100

z = (226 - 50)/5

z = 35.2

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A fair coin is tossed until 100 heads appear. Find the probability that at least 226 tosses will be necessary.

Summary: 

The probability that at least 226 tosses will be necessary when a fair coin is tossed is a normal distribution and P(until 100 heads appear) is 35.2