QuestionFind the numbers for a Venn diagram where $P(B|A) = \frac{8}{24}$ , given 36 flu cases and 24 flu shots.

Studdy Solution

STEP 1

Assumptions1. Event $A$ represents those who received a flu shot. Event $B$ represents those who caught the flu3. The probability that a person catches the flu given that they've had a flu shot is $\frac{8}{24}$
4. We know that36 people caught the flu and24 people received flu shots

STEP 2

First, we need to find the number of people who caught the flu and had a flu shot. We can do this by multiplying the probability that a person catches the flu given that they've had a flu shot by the total number of people who had a flu shot.
$Number\, of\, people\, who\, caught\, the\, flu\, and\, had\, a\, flu\, shot = Probability\, of\, catching\, the\, flu\, given\, a\, flu\, shot \\times Total\, number\, of\, people\, who\, had\, a\, flu\, shot$

STEP 3

Now, plug in the given values for the probability and the total number of people who had a flu shot to calculate the number of people who caught the flu and had a flu shot.
$Number\, of\, people\, who\, caught\, the\, flu\, and\, had\, a\, flu\, shot = \frac{8}{24} \\times24$

STEP 4

Calculate the number of people who caught the flu and had a flu shot.
$Number\, of\, people\, who\, caught\, the\, flu\, and\, had\, a\, flu\, shot = \frac{8}{24} \\times24 =8$

STEP 5

Now that we have the number of people who caught the flu and had a flu shot, we can find the number of people who had a flu shot but did not catch the flu. This can be found by subtracting the number of people who caught the flu and had a flu shot from the total number of people who had a flu shot.
$Number\, of\, people\, who\, had\, a\, flu\, shot\, but\, did\, not\, catch\, the\, flu = Total\, number\, of\, people\, who\, had\, a\, flu\, shot - Number\, of\, people\, who\, caught\, the\, flu\, and\, had\, a\, flu\, shot$

STEP 6

Plug in the values for the total number of people who had a flu shot and the number of people who caught the flu and had a flu shot.
$Number\, of\, people\, who\, had\, a\, flu\, shot\, but\, did\, not\, catch\, the\, flu =24 -8$

STEP 7

Calculate the number of people who had a flu shot but did not catch the flu.
$Number\, of\, people\, who\, had\, a\, flu\, shot\, but\, did\, not\, catch\, the\, flu =24 - =16$

STEP 8

Now that we have the number of people who had a flu shot but did not catch the flu, we can find the number of people who caught the flu but did not have a flu shot. This can be found by subtracting the number of people who caught the flu and had a flu shot from the total number of people who caught the flu.
$Number\, of\, people\, who\, caught\, the\, flu\, but\, did\, not\, have\, a\, flu\, shot = Total\, number\, of\, people\, who\, caught\, the\, flu - Number\, of\, people\, who\, caught\, the\, flu\, and\, had\, a\, flu\, shot$

STEP 9

Plug in the values for the total number of people who caught the flu and the number of people who caught the flu and had a flu shot.
$Number\, of\, people\, who\, caught\, the\, flu\, but\, did\, not\, have\, a\, flu\, shot =36 -8$

STEP 10

Calculate the number of people who caught the flu but did not have a flu shot.
$Number\, of\, people\, who\, caught\, the\, flu\, but\, did\, not\, have\, a\, flu\, shot =36 -8 =28$ So, the Venn diagram should be filled as follows- The intersection of Event $A$ and Event $B$ (those who received a flu shot and caught the flu) is8. - The number of people who received a flu shot but did not catch the flu (Event $A$ excluding the intersection) is16. - The number of people who caught the flu but did not receive a flu shot (Event $B$ excluding the intersection) is28.

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