Question

in the first episode of a reality show, contestants had to spin two wheels of fate. spinning the first wheel determined the remote location where contestants would reside for the duration of the season. spinning the second wheel determined which \bonus survival tool\ they would be allowed to bring, along with a few other necessary items. the probability that a participant spun the first wheel and landed on desert is 0.6, and the probability that a participant spun the second wheel and landed on matches is 0.7. what is the probability that a randomly chosen participant spun the first wheel and landed on desert or spun the second wheel and landed on matches if those are independent events? write your answer as a whole number, decimal, or simplified fraction.

Explanation:

Step1: Recall the formula for the probability of the union of two independent events

For two independent events \( A \) and \( B \), the probability of \( A \) or \( B \) (denoted as \( P(A \cup B) \)) is given by the formula:
\( P(A \cup B) = P(A) + P(B) - P(A \cap B) \).
Since \( A \) and \( B \) are independent, \( P(A \cap B) = P(A) \times P(B) \).

Let \( A \) be the event "spun the first wheel and landed on desert" with \( P(A) = 0.6 \), and \( B \) be the event "spun the second wheel and landed on matches" with \( P(B) = 0.7 \).

Step2: Calculate \( P(A \cap B) \)

Because \( A \) and \( B \) are independent,
\( P(A \cap B) = P(A) \times P(B) = 0.6 \times 0.7 = 0.42 \).

Step3: Calculate \( P(A \cup B) \)

Substitute \( P(A) = 0.6 \), \( P(B) = 0.7 \), and \( P(A \cap B) = 0.42 \) into the union formula:
\( P(A \cup B) = 0.6 + 0.7 - 0.42 \).

First, add \( 0.6 \) and \( 0.7 \): \( 0.6 + 0.7 = 1.3 \).
Then subtract \( 0.42 \): \( 1.3 - 0.42 = 0.88 \).

Answer:

\( 0.88 \)