Chapter 10.3, Problem 11Q

To determine

the probability that a randomly selected runner who achieves the goal time.

Answer to Problem 11Q

	Own Team	Home Team	Rival Team	Total
Achieve goal	$9$	$6$	$8$	$0.48\left(23/48\right)$
Does not achieve goal	$6$	$14$	$5$	$0.52\left(1-0.48\right)$
Total	$0.31\left(15/48\right)$	$0.42\left(20/48\right)$	$0.27\left(13/48\right)$	$1$

Therefore, the probability that a randomly selected runner who achieves his/her goal time is from own team is $0.19$ .

Explanation of Solution

Given:

Of the 15, 9 achieve their goal times.

Of the 20 athletes, 6 achieve their goal times.

On your rival’s team, 8 of the 13 athletes achieve their goal times.

Calculation:

There are $48$ athletes in total. Divide the row and column totals by the total number of athletes. This gives the marginal frequencies for the row and column. The following table shows the results.

	Own Team	Home Team	Rival Team	Total
Achieve goal	$9$	$6$	$8$	$0.48\left(23/48\right)$
Does not achieve goal	$6$	$14$	$5$	$0.52\left(1-0.48\right)$
Total	$0.31\left(15/48\right)$	$0.42\left(20/48\right)$	$0.27\left(13/48\right)$	$1$

Divide the frequency in the cell (Achieve goal-own team) by the total number of people surveyed. This gives the probability that a randomly selected runner who achieves his/her goal time is from own team. The required probability is as follows:

  $\begin{array}{l}P\left(achievegoal-ownteam\right)=\frac{9}{48}\\ \text{ =0}\text{.19}\end{array}$

Therefore, the answer is: $\boxed{0.19}$ .

Conclusion:

	Own Team	Home Team	Rival Team	Total
Achieve goal	$9$	$6$	$8$	$0.48\left(23/48\right)$
Does not achieve goal	$6$	$14$	$5$	$0.52\left(1-0.48\right)$
Total	$0.31\left(15/48\right)$	$0.42\left(20/48\right)$	$0.27\left(13/48\right)$	$1$

Therefore, the probability that a randomly selected runner who achieves his/her goal time is from own team is $0.19$ .