Chapter 13.5, Problem 8CFU

To determine

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

Answer to Problem 8CFU

The probability all heads turn up when three coins are tossed provided that at least two coins shows heads is $\frac{1}{4}$ .

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 two coins shows heads.

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

Therefore, outcomes of $B=\left\{HHH,HTH,HHT,THH\right\}$ 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 two coins shows head,

  $P\left(B\right)=\frac{4}{8}$

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 two coins shows heads.

  $\begin{array}{c}P\left(A|B\right)=\frac{\frac{1}{8}}{\frac{4}{8}}\\ =\frac{1}{4}\end{array}$

Thus, the probability all heads turn up when three coins are tossed provided that at least two coins shows headsis $\frac{1}{4}$ .