#### Question

While he was a prisoner of war during World War II, John Kerrich tossed a coin 10,000 times. He got 5067 heads. If the coin is perfectly balanced, the probability of a head is 0.5. Is there reason to think that Kerrich's coin was not balanced? To answer this question, use a Normal distribution to estimate the probability that tossing a balanced coin 10,000 times would give a count of heads at least this far from 5000 (that is, at least 5067 heads or at most 4933 heads.)

#### Solution

Verified#### Step 1

1 of 3Given:

$n=10000$

$p=0.5$

The mean of a binomial variable is the product of the sample size $n$ and the probability $p$:

$\mu_X=np=10000(0.5)=5000$

The standard deviation of a binomial variable is the square root of the product of the sample size $n$, the probability $p$ and the probability $1-p$:

$\sigma_X=\sqrt{np(1-p)}=\sqrt{10000(0.5)(1-0.5)}=50$

The z-score is the value decreased by the mean divided by the standard deviation:

$z=\dfrac{x-\mu}{\sigma}=\dfrac{4933-5000}{50}\approx -1.34$

$z=\dfrac{x-\mu}{\sigma}=\dfrac{5067-5000}{50}\approx 1.34$

Determine the corresponding probability using table A:

$P(X\leq 4933\text{ or } X\geq 5067)=P(Z\leq -1.34\text{ or } X\geq 1.34)=2\times P(Z<-1.34)$

$=2\times 0.0901=0.1802=18.02\%$

There is no reason to think that Kerrich's coin was not balanced, because the probability is more than 5% and thus the event is likely to occur.