Chapter 13.5, Problem 7CFU

To determine

To calculate: The probability all heads turn up when three coins are tossed provided that at least one coin shows a head.

Answer to Problem 7CFU

The probability all heads turn up when three coins are tossed provided that at least one coin shows a headis $\frac{1}{7}$ .

Explanation of Solution

Given information:

Three coins are tossed.

Formula used:

Probability of an event Eis $P\left(\text{E}\right)=\frac{\text{number of outcome}}{\text{total number of outcome}}$ .

Given an event Q , the conditional probability of event A is $P\left(A|Q\right)=\frac{P\left(A\text{ and Q}\right)}{P\left(Q\right)}$ where $P\left(Q\right)\ne 0$ .

Calculation:

Consider the provided information thatthree coins are tossed.

Sample space of the event is $S=\left\{HHH,HTT,HTH,HHT,TTT,THH,THT,TTH\right\}$

The probability all heads turn up when three coins are tossed provided that at least one coin shows a head.

Let A be an event that all heads turn up and B be an event that at least one shows head.

Therefore, outcomes of $B=\left\{HHH,HTT,HTH,HHT,THH,THT,TTH\right\}$ and outcomes of $\text{A and B}=\left\{HHH\right\}$

Recall that probability of an event Eis $P\left(\text{E}\right)=\frac{\text{number of outcome}}{\text{total number of outcome}}$ .

Probability that at least one shows head,

  $P\left(B\right)=\frac{7}{8}$

Probability that all heads and first coin shows head,

  $P\left(A\text{ and B}\right)=\frac{1}{8}$

Recall that given an event Q , the conditional probability of event A is $P\left(A|Q\right)=\frac{P\left(A\text{ and Q}\right)}{P\left(Q\right)}$ where $P\left(Q\right)\ne 0$ .

The probability all heads turn up when three coins are tossed provided that at least one coin shows a head.

  $\begin{array}{c}P\left(A|B\right)=\frac{\frac{1}{8}}{\frac{7}{8}}\\ =\frac{1}{7}\end{array}$

Thus, the probability all heads turn up when three coins are tossed provided that at least one coin shows a head is $\frac{1}{7}$ .