Question

If this procedure is repeated 100 times, what is the probability that the number of times that the coin lands tails will be less than 40

Answer 1

1.79% probability that the number of times that the coin lands tails will be less than 40

Explanation:

Binomial probability distribution

Probability of exactly x sucesses on n repeated trials, with p probability.

Can be approximated to a normal distribution, using the expected value and the standard deviation.

The expected value of the binomial distribution is:

`E(X) = np`

The standard deviation of the binomial distribution is:

`√(V(X)) = √(np(1-p))`

Normal probability distribution

Problems of normally distributed samples can be solved using the z-score formula.

In a set with mean
`\mu` and standard deviation
`\sigma`, the zscore of a measure X is given by:

`Z = (X - \mu)/(\sigma)`

The Z-score measures how many standard deviations the measure is from the mean. After finding the Z-score, we look at the z-score table and find the p-value associated with this z-score. This p-value is the probability that the value of the measure is smaller than X, that is, the percentile of X. Subtracting 1 by the pvalue, we get the probability that the value of the measure is greater than X.

When we are approximating a binomial distribution to a normal one, we have that
`\mu = E(X)`,
`\sigma = √(V(X))`.

Fair coin:

Equally as likely to be heads or tails, so
`p = 0.5`

100 times

`n = 100`

Then

`\mu = E(X) = np = 100*0.5 = 50`

`\sigma = √(V(X)) = √(np(1-p)) = √(100*0.5*0.5) = 5`

What is the probability that the number of times that the coin lands tails will be less than 40

Using continuity correction, this is
`P(X < 40 - 0.5) = P(X < 39.5)`, which is the pvalue of Z when X = 39.5.

`Z = (X - \mu)/(\sigma)`

`Z = (39.5 - 50)/(5)`

`Z = -2.1`

`Z = -2.1` has a pvalue of 0.0179

1.79% probability that the number of times that the coin lands tails will be less than 40