Question

In a group of 25 boys, 20 play ice hockey and 17 play baseball. They all play at least one of the games.

What is the probability that a boy chosen at random from the class plays ice hockey but not baseball?

Answer 1

To find the probability of a boy playing ice hockey but not baseball in a group where 20 play ice hockey, 17 play baseball, and all play at least one game, we subtract the number who play both sports (calculated using the inclusion-exclusion principle) from those who play ice hockey. There are 8 boys playing only ice hockey, so the probability is 8/25 or 32%.

Step-by-step explanation:

The question is about probability in the context of students playing different sports. To find the probability that a boy chosen at random from a class of 25 (where 20 play ice hockey and 17 play baseball, and all play at least one of the games) plays ice hockey but not baseball, we use the principle of inclusion and exclusion for two sets.

The total number of boys playing ice hockey is given as 20. The number playing baseball is 17. The principle of inclusion and exclusion states that to find the number who play only ice hockey, we subtract from the total ice hockey players the number who play both sports. Let's call the number playing both sports 'x'.

The number of students playing at least one sport is 25 (given all play at least one of the games). Therefore, using the inclusion-exclusion principle:

1. Total playing at least one sport = Number playing ice hockey + Number playing baseball - Number playing both sports
2. 25 = 20 + 17 - x
3. x = 20 + 17 - 25
4. x = 12

This means that 12 boys play both ice hockey and baseball. To find those who play only ice hockey, we subtract those who play both from those who play ice hockey:

1. Number playing only ice hockey = Total playing ice hockey - Number playing both sports
2. Number playing only ice hockey = 20 - 12
3. Number playing only ice hockey = 8

Now that we know there are 8 boys who play only ice hockey, to find the probability that a randomly chosen boy plays ice hockey but not baseball, we divide this number by the total number of boys.

Probability = Number playing only ice hockey / Total number of boys

Probability = 8 / 25

The probability is 8/25 or 0.32 (which can also be expressed as 32%).