a First, let's add the row and column totals to the table.
Gender |
Clubs |
No Clubs |
Total
|
Male |
156 |
242 |
398
|
Female |
312 |
108 |
420
|
Total |
468 |
350 |
818
|
From the bottom right entry, we can see that there were
818 students in King High School. If we divide the numbers with this total, we will get the probabilities associated with the cells. Let's see the probability that a randomly selected student is a male club member.
P(male and club member)=818156
P(male and club member)=40978
The calculation of the other entries is similar. Let's write all the probabilities in fully simplified form.
Gender |
Clubs |
No Clubs |
Total
|
male |
40978 |
409121 |
409199
|
female |
409156 |
40954 |
409210
|
Total |
409234 |
409175 |
1
|
We are interested in the probability that a randomly selected student is a member of a club,
given that he is male.
Let's use the to write this probability as the quotient of two unconditional probabilities.
P(club member∣male)=P(male)P(club member and male)
Let's use the color we used in the formula to highlight the relevant entries of the table.
Gender |
Clubs |
No Clubs |
Total
|
Male |
40978 |
409121 |
409199
|
Female |
409156 |
40954 |
409210
|
Total |
409234 |
409175 |
1
|
If we substitute these values in the formula, we get the answer to the question.
P(club member∣male)=199/40978/409=19978≈0.392
The probability that a randomly selected student is a member of a club, given that he is male, is
19978, or about
0.392.