Questions: Use a Venn diagram. Let P(Z)=0.47, P(Y)=0.34, and P(Z ∪ Y)=0.63. Find each probability. (a) P(Z' ∩ Y') (b) P(Z' ∪ Y') (c) P(Z' ∪ Y) (d) P(Z ∩ Y') Complete the Venn diagram using the given probabilities. region I represents 0.29 region II represents 0.18 region III represents 0.16 region IV represents 0.37 (Type integers or decimals.) (a) P(Z' ∩ Y')=0.37 (Type an integer or a decimal.) (b) P(Z' ∪ Y')=0.82 (Type an integer or a decimal.) (c) P(Z' ∪ Y)=0.71 (Type an integer or a decimal.) (d) P(Z ∩ Y')=

Use a Venn diagram. Let P(Z)=0.47, P(Y)=0.34, and P(Z ∪ Y)=0.63. Find each probability.
(a) P(Z' ∩ Y')
(b) P(Z' ∪ Y')
(c) P(Z' ∪ Y)
(d) P(Z ∩ Y')

Complete the Venn diagram using the given probabilities.
region I represents 0.29
region II represents 0.18
region III represents 0.16
region IV represents 0.37
(Type integers or decimals.)
(a) P(Z' ∩ Y')=0.37
(Type an integer or a decimal.)
(b) P(Z' ∪ Y')=0.82
(Type an integer or a decimal.)
(c) P(Z' ∪ Y)=0.71
(Type an integer or a decimal.)
(d) P(Z ∩ Y')=

Transcript text: Part 5 of 5 Use a Venn diagram. Let $P(Z)=0.47, P(Y)=0.34$, and $P(Z \cup Y)=0.63$. Find each probability. (a) $P\left(Z^{\prime} \cap Y^{\prime}\right)$ (b) $P\left(Z^{\prime} \cup Y^{\prime}\right)$ (c) $P\left(Z^{\prime} \cup Y\right)$ (d) $P\left(Z \cap Y^{\prime}\right)$ Complete the Venn diagram using the given probabilities. region I represents 0.29 region II represents 0.18 region III represents 0.16 region IV represents 0.37 (Type integers or decimals.) (a) $P\left(Z^{\prime} \cap Y^{\prime}\right)=0.37$ (Type an integer or a decimal.) (b) $P\left(Z^{\prime} \cup Y^{\prime}\right)=0.82$ (Type an integer or a decimal.) (c) $P\left(Z^{\prime} \cup Y\right)=0.71$ (Type an integer or a decimal.) (d) $P\left(Z \cap Y^{\prime}\right)=$ $\square$ (Type an integer or a decimal.)

Solution

Solution Steps

Step 1: Identify Given Probabilities

Given:

$ P(Z) = 0.47 $
$ P(Y) = 0.34 $
$ P(Z \cup Y) = 0.63 $

Step 2: Calculate $ P(Z \cap Y) $

Using the formula for the union of two events: \[ P(Z \cup Y) = P(Z) + P(Y) - P(Z \cap Y) \] \[ 0.63 = 0.47 + 0.34 - P(Z \cap Y) \] \[ P(Z \cap Y) = 0.81 - 0.63 \] \[ P(Z \cap Y) = 0.18 \]

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

Using the complement rule: \[ P(Z' \cap Y') = 1 - P(Z \cup Y) \] \[ P(Z' \cap Y') = 1 - 0.63 \] \[ P(Z' \cap Y') = 0.37 \]

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

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

Step 5: Calculate $ P(Z' \cup Y) $

Using the complement rule and the inclusion-exclusion principle: \[ P(Z' \cup Y) = 1 - P(Z \cap Y') \] \[ P(Z \cap Y') = P(Z) - P(Z \cap Y) \] \[ P(Z \cap Y') = 0.47 - 0.18 \] \[ P(Z \cap Y') = 0.29 \] \[ P(Z' \cup Y) = 1 - 0.29 \] \[ P(Z' \cup Y) = 0.71 \]

Final Answer

(a) $ P(Z' \cap Y') = 0.37 $

(b) $ P(Z' \cup Y') = 0.82 $

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