Question

based on the data in this two - way table, which statement is true?

type of flower/color	red	pink	yellow	total
hibiscus	80	40	90	210
total	120	60	135	315

a. $p(\text{flower is yellow}|\text{flower is rose})\
eq p(\text{flower is yellow})$
b. $p(\text{flower is hibiscus}|\text{color is red}) = p(\text{flower is hibiscus})$
c. $p(\text{flower is rose}|\text{color is red}) = p(\text{flower is red})$
d. $p(\text{flower is hibiscus}|\text{color is pink})\
eq p(\text{flower is hibiscus})$

Explanation:

Step1: Recall conditional probability formula

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

Step2: Calculate for Option A

Find $P(\text{yellow}|\text{rose})=\frac{45}{105}=\frac{3}{7}$, $P(\text{yellow})=\frac{135}{315}=\frac{3}{7}$. So $\frac{3}{7}=\frac{3}{7}$, A is false.

Step3: Calculate for Option B

Find $P(\text{hibiscus}|\text{red})=\frac{80}{120}=\frac{2}{3}$, $P(\text{hibiscus})=\frac{210}{315}=\frac{2}{3}$. So $\frac{2}{3}=\frac{2}{3}$, B is true.

Step4: Verify remaining options (optional)

For C: $P(\text{rose}|\text{red})=\frac{40}{120}=\frac{1}{3}$, $P(\text{red})=\frac{120}{315}=\frac{8}{21}$. $\frac{1}{3}
eq\frac{8}{21}$, C is false.
For D: $P(\text{hibiscus}|\text{pink})=\frac{40}{60}=\frac{2}{3}$, $P(\text{hibiscus})=\frac{210}{315}=\frac{2}{3}$. $\frac{2}{3}=\frac{2}{3}$, D is false.

Answer:

B. P(flower is hibiscus|color is red) = P(flower is hibiscus)