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| | | |large|medium|total|red|42|48|90|blue|35|40|75|total|77|88|165| from the data given in the table, we can infer that p(blue shirt | large shirt) = p(blue shirt) p(large shirt | red shirt) = p(red shirt) p(shirt is medium and blue) = p(medium shirt) p(red shirt | large shirt) = p(large shirt)

Explanation:

Step1: Recall conditional - probability formula

The formula for conditional probability is $P(A|B)=\frac{P(A\cap B)}{P(B)}$.

Step2: Calculate $P(\text{blue shirt}|\text{large shirt})$

$P(\text{blue shirt}|\text{large shirt})=\frac{n(\text{blue and large})}{n(\text{large})}=\frac{35}{77}=\frac{5}{11}$. And $P(\text{blue shirt})=\frac{n(\text{blue})}{n(\text{total})}=\frac{75}{165}=\frac{5}{11}$. So $P(\text{blue shirt}|\text{large shirt}) = P(\text{blue shirt})$.

Step3: Calculate $P(\text{large shirt}|\text{red shirt})$

$P(\text{large shirt}|\text{red shirt})=\frac{n(\text{large and red})}{n(\text{red})}=\frac{42}{90}=\frac{7}{15}$, and $P(\text{red shirt})=\frac{90}{165}=\frac{6}{11}$. Since $\frac{7}{15}
eq\frac{6}{11}$, $P(\text{large shirt}|\text{red shirt})
eq P(\text{red shirt})$.

Step4: Calculate $P(\text{medium and blue})$

$P(\text{medium and blue})=\frac{n(\text{medium and blue})}{n(\text{total})}=\frac{40}{165}=\frac{8}{33}$, and $P(\text{medium shirt})=\frac{88}{165}$. Since $\frac{8}{33}
eq\frac{88}{165}$, $P(\text{medium and blue})
eq P(\text{medium shirt})$.

Step5: Calculate $P(\text{red shirt}|\text{large shirt})$

$P(\text{red shirt}|\text{large shirt})=\frac{n(\text{red and large})}{n(\text{large})}=\frac{42}{77}=\frac{6}{11}$, and $P(\text{large shirt})=\frac{77}{165}$. Since $\frac{6}{11}
eq\frac{77}{165}$, $P(\text{red shirt}|\text{large shirt})
eq P(\text{large shirt})$.

Answer:

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