Question

e) find the standard deviation for the number of heads.
round final answer to 3 decimal places. write your units in the second box.
0.866 heads
which of the following is the correct symbol for the standard deviation of a probability distribution? σ
f) find the probability of having two or less heads.

Explanation:

Step1: Assume it is a binomial distribution

Let's assume this is a binomial distribution \(X\sim B(n,p)\). For a fair coin, \(n = 3\) (number of trials, assume 3 - coin - toss situation as a common case if not specified otherwise) and \(p=\frac{1}{2}\). The probability mass function of a binomial distribution is \(P(X = k)=C(n,k)p^{k}(1 - p)^{n - k}\), where \(C(n,k)=\frac{n!}{k!(n - k)!}\)

Step2: Calculate \(P(X\leq2)\)

We know that \(P(X\leq2)=1 - P(X = 3)\)
For \(k = 3\), \(n = 3\), \(p=\frac{1}{2}\)
\(P(X = 3)=C(3,3)(\frac{1}{2})^{3}(1-\frac{1}{2})^{3 - 3}\)
Since \(C(3,3)=\frac{3!}{3!(3 - 3)!}=1\)
\(P(X = 3)=1\times(\frac{1}{2})^{3}\times1=\frac{1}{8}=0.125\)
Then \(P(X\leq2)=1 - 0.125 = 0.875\)

Answer:

\(0.875\)