(part 3)

c. This probability question is a conditional. You are asked to find the probability that an eight-week-old baby smiles more than 12 seconds when you already know the baby has smiled for more than eight seconds.

Find P(x > 12|x > 8) There are two ways to do the problem. For the first way, use the fact that this is a conditional and changes the sample space. The graph illustrates the new sample space. You already know the baby smiled more than eight seconds.

for 8 < x < 23

Write a new f(x): f(x) = $\frac{1}{23-\text{ 8}}$ = $\frac{1}{15}$ for 8 < x < 23.

P(x > 12|x > 8) = (23 − 12)$\left(\frac{1}{15}\right)$ = $\frac{11}{15}$

Figure 5.13

For the second way, use the conditional formula from Probability Topics with the original distribution.

P(A|B) = $\frac{P(A\text{ AND }B)}{P(B)}$

For this problem, A is (x > 12) and B is (x > 8).

So, P(x > 12|x > 8) = $\frac{(x>12\text{ AND }x>8)}{P(x>8)}=\frac{P(x>12)}{P(x>8)}=\frac{\frac{11}{23}}{\frac{15}{23}}=\frac{11}{15}$

Figure 5.14