Chapter 13.5, Problem 14E

To determine

To calculate: The probability one coin shows heads when two coins are tossed provided that at least one coin is tails.

Answer to Problem 14E

The probability one coin shows heads when two coins are tossed provided that at least one coin is tails is $\frac{2}{3}$ .

Explanation of Solution

Given information:

Two 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 thattwo coins are tossed.

Sample space of the event is $S=\left\{HH,TT,HT,TH\right\}$

The probability one coin shows heads when two coins are tossed provided that at least one coin is tails.

Let A be an event that one coin shows head and B be an event that at least one coin is tails.

Therefore, outcomes of $B=\left\{TT,TH,HT\right\}$ and outcomes of $\text{A and B}=\left\{TH,HT\right\}$

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 coin is tail,

  $P\left(B\right)=\frac{3}{4}$

Probability that one coin shows heads and at least one coin is tails,

  $P\left(A\text{ and B}\right)=\frac{2}{4}$

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 one coin shows heads when two coins are tossed provided that at least one coin is tails.

  $\begin{array}{c}P\left(A|B\right)=\frac{\frac{2}{4}}{\frac{3}{4}}\\ =\frac{2}{3}\end{array}$

Thus, the probability one coin shows heads when two coins are tossed provided that at least one coin is tails is $\frac{2}{3}$ .