Question

select the correct answer from the drop-down menu.
a box contains shirts in two different colors and two different sizes. the numbers of shirts of each color and size are given in the table.

shirt color	size
red	42	48	90
blue	35	40	75
total	77	88	165

from the data given in the table, we can infer tha
p(red shirt | large shirt) = p(large shirt)
p(shirt is medium and blue) = p(medium shirt)
p(large shirt | red shirt) = p(red shirt)
p(blue shirt | large shirt) = p(blue shirt)

Explanation:

Step1: Recall conditional probability rule

For events A and B, $P(A|B)=\frac{P(A\cap B)}{P(B)}$. Also, $P(A|B)=P(A)$ if A and B are independent.

Step2: Calculate for each option

Option1: $P(\text{red shirt} | \text{large shirt}) = P(\text{large shirt})$

$P(\text{red shirt} | \text{large shirt})=\frac{42}{77}\approx0.545$, $P(\text{large shirt})=\frac{77}{165}\approx0.467$. Not equal.

Option2: $P(\text{medium and blue}) = P(\text{medium shirt})$

$P(\text{medium and blue})=\frac{40}{165}\approx0.242$, $P(\text{medium shirt})=\frac{88}{165}\approx0.533$. Not equal.

Option3: $P(\text{large shirt} | \text{red shirt}) = P(\text{red shirt})$

$P(\text{large shirt} | \text{red shirt})=\frac{42}{90}\approx0.467$, $P(\text{red shirt})=\frac{90}{165}\approx0.545$. Not equal.

Option4: $P(\text{blue shirt} | \text{large shirt}) = P(\text{blue shirt})$

$P(\text{blue shirt} | \text{large shirt})=\frac{35}{77}=\frac{5}{11}\approx0.455$, $P(\text{blue shirt})=\frac{75}{165}=\frac{5}{11}\approx0.455$. These are equal.

Answer:

P(blue shirt | large shirt) = P(blue shirt)