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') $: \[ P(Z' \cap Y') = 1 - 0.74 = 0.26 \]

Step 3: Calculate $ P(Z' \cup Y') $

Using De Morgan's law: \[ P(Z' \cup Y') = 1 - P(Z \cap Y) \] \[ P(Z' \cup Y') = 1 - 0.18 = 0.82 \]

Step 4: Calculate $ P(Z' \cap Y) $

Using the formula for the complement of the intersection: \[ P(Z' \cap Y) = P(Y) - P(Z \cap Y) \] \[ P(Z' \cap Y) = 0.42 - 0.18 = 0.24 \]