Question

b) write the probability distribution for the number of heads. make sure to put the values of x from smallest to largest.

Explanation:

1. Assume the number of coin - tosses is \(n\). Here, let's first consider the case of \(n = 3\) coin - tosses (a common example when not specified otherwise). The number of heads \(X\) can take values \(0\), \(1\), \(2\), \(3\).

• The probability of getting \(k\) heads in \(n\) independent and identical coin - tosses is given by the binomial probability formula \(P(X = k)=C(n,k)\times p^{k}\times(1 - p)^{n - k}\), where \(C(n,k)=\frac{n!}{k!(n - k)!}\) is the number of combinations of \(n\) items taken \(k\) at a time, and \(p\) is the probability of getting a head in a single coin - toss. For a fair coin, \(p=\frac{1}{2}\).

1. When \(X = 0\) (no heads in \(n = 3\) tosses):

• Calculate the combination \(C(3,0)=\frac{3!}{0!(3 - 0)!}=\frac{3!}{3!}=1\).
• Using the binomial formula \(P(X = 0)=C(3,0)\times(\frac{1}{2})^{0}\times(1-\frac{1}{2})^{3 - 0}=1\times1\times(\frac{1}{2})^{3}=\frac{1}{8}\).

1. When \(X = 1\) (one head in 3 tosses):

• Calculate the combination \(C(3,1)=\frac{3!}{1!(3 - 1)!}=\frac{3!}{1!2!}=\frac{3\times2!}{2!}=3\).
• Using the binomial formula \(P(X = 1)=C(3,1)\times(\frac{1}{2})^{1}\times(1 - \frac{1}{2})^{3 - 1}=3\times\frac{1}{2}\times(\frac{1}{2})^{2}=\frac{3}{8}\).

1. When \(X = 2\) (two heads in 3 tosses):

• Calculate the combination \(C(3,2)=\frac{3!}{2!(3 - 2)!}=\frac{3!}{2!1!}=\frac{3\times2!}{2!}=3\).
• Using the binomial formula \(P(X = 2)=C(3,2)\times(\frac{1}{2})^{2}\times(1 - \frac{1}{2})^{3 - 2}=3\times(\frac{1}{2})^{2}\times\frac{1}{2}=\frac{3}{8}\).

1. When \(X = 3\) (three heads in 3 tosses):

• Calculate the combination \(C(3,3)=\frac{3!}{3!(3 - 3)!}=\frac{3!}{3!0!}=1\).
• Using the binomial formula \(P(X = 3)=C(3,3)\times(\frac{1}{2})^{3}\times(1 - \frac{1}{2})^{3 - 3}=1\times(\frac{1}{2})^{3}\times1=\frac{1}{8}\).

Step1: Define possible values of \(X\)

The number of heads \(X\) in coin - tosses can be \(0,1,2,\cdots,n\). For simplicity, assume \(n = 3\).

Step2: Calculate \(P(X = 0)\)

Use binomial formula with \(C(3,0) = 1\), \(p=\frac{1}{2}\), \(n = 3\), \(k = 0\).

Step3: Calculate \(P(X = 1)\)

Find \(C(3,1)=3\) and use binomial formula.

Step4: Calculate \(P(X = 2)\)

Find \(C(3,2)=3\) and use binomial formula.

Step5: Calculate \(P(X = 3)\)

Find \(C(3,3)=1\) and use binomial formula.

Answer:

\(X\)	\(P(X)\)
\(1\)	\(\frac{3}{8}\)
\(2\)	\(\frac{3}{8}\)
\(3\)	\(\frac{1}{8}\)