Questions: Let P(Z)=0.50, P(Y)=0.42, and P(Z ∩ Y)=0.18. Use a Venn diagram to find (a) P(Z′ ∩ Y′), (b) P(Z′ ∪ Y′), (c) P(Z′ ∪ Y), and (d) P(Z ∩ Y′). (a) P(Z′ ∩ Y′)=0.26 (b) P(Z′ ∪ Y′)=0.82 (c) P(Z′ ∪ Y)=0.68 (d) P(Z ∩ Y′)=

Transcript text: Let $P(Z)=0.50, P(Y)=0.42$, and $P(Z \cap Y)=0.18$. Use a Venn diagram to find (a) $P\left(Z^{\prime} \cap Y^{\prime}\right)$, (b) $P\left(Z^{\prime} \cup Y^{\prime}\right)$, (c) $P(Z^{\prime} \cup Y)$, and (d) $P\left(Z \cap Y^{\prime}\right)$. (a) $P\left(Z^{\prime} \cap Y^{\prime}\right)=0.26$ (b) $\mathrm{P}\left(\mathrm{Z}^{\prime} \cup Y^{\prime}\right)=0.82$ (c) $P\left(Z^{\prime} \cup Y\right)=0.68$ (d) $P\left(Z \cap Y^{\prime}\right)=$

Solution

Solution Steps

Step 1: Identify Given Probabilities

Given:

$P(Z) = 0.50$
$P(Y) = 0.42$
$P(Z \cap Y) = 0.18$

Step 2: Calculate

P(Z' \cap Y')

Using the formula for the complement of the union: $P(Z' \cap Y') = 1 - P(Z \cup Y)$ First, find $P(Z \cup Y)$ using the formula: $P(Z \cup Y) = P(Z) + P(Y) - P(Z \cap Y)$ $P(Z \cup Y) = 0.50 + 0.42 - 0.18 = 0.74$ Now, calculate $P(Z' \cap Y')$ : $P(Z' \cap Y') = 1 - 0.74 = 0.26$