Question

There are 16 kids on a soccer team. The players are either boys or girls, and they either prefer playing offense or defense. There are 6 boys who prefer offense, 3 boys who prefer defense, 5 girls who prefer offense, and 2 girls who prefer defense.

Suppose you choose one player from the team at random. Let event A be that the player is a girl, and event B be that the player prefers defense. Fine P(B | A).

Answer 1

To determine P(B | A), the probability that a player prefers defense given that the player is a girl, divide the number of girls who prefer defense (2) by the total number of girls (7), which gives us 0.2857 or 28.57%.

Step-by-step explanation:

You have asked to find P(B | A), the probability that a randomly chosen player prefers defense, given that the player is a girl. There are 7 girls on the soccer team based on the information provided: 5 girls who prefer offense and 2 girls who prefer defense. The total number of girls who prefer defense is therefore 2.

P(B | A) is the probability of event B occurring given that event A has occurred. To calculate this, we need to divide the number of favorable outcomes for both A and B by the total number of outcomes in event A. In this context, event A is that the player is a girl, and event B is that the player prefers defense. Since there are 2 girls preferring defense out of the total 7 girls, the calculation for P(B | A) is:

P(B | A) = Number of girls preferring defense / Total number of girls
P(B | A) = 2 / 7

So, the probability is approximately 0.2857 or 28.57%.

Answer 2

`P(B|A)=(2)/(7)`

Step-by-step explanation:

The probability of
`P(B|A)` can be read as the probability of event B occurring given event A. In this question, event A occurs when the chosen player is a girl. There are 7 girls on the soccer team. Event B occurs when the chose player plays defense. Since
`P(B|A)` stipulates that event A already occurred, we want the probability of choosing a player who prefers defense from the 7 girls. There are 2 girls who prefer defense, hence
`P(B|A)=\boxed{(2)/(7)}`.

Alternative:

For dependent events
`A` and
`B`, the conditional probability of event B occurring given A is given by:

`P(B|A)=P(B\cap A)/ P(A)`

`P(B\cap A)` indicates the intersection of
`P(B)` and
`P(A)`. In this case, it is the probability that both events occur. Since there are 16 kids on the soccer team and only 2 are girls and prefer defense,
`P(B\cap A)=(2)/(16)=(1)/(8)`. The probability of event A occurring (chosen player is a girl) is equal to the number of girls (7) divided by the number of kids on the team (16), hence
`P(A)=(7)/(16)`.

Therefore, the probability of event B occurring, given event A occurred, is equal to:

`P(B|A)=(1)/(8)/ (7)/(16),\\\\P(B|A)=(1)/(8)\cdot (16)/(7)=\boxed{(2)/(7)}`