Questions: Rickey, Ashley and Loretta decide to enter a marathon race. The respective probabilities that they will complete the race is 0.9, 0.7 and 0.6. a) How many total outcomes are possible? Draw a tree diagram. b) Assuming that their performances are independent, find the probability that: i) they all complete the race. ii) at most one complete the race. iii) at least two complete the race.

Solution Steps

a) To find the total number of outcomes, consider each person either completing or not completing the race. Since there are three participants, each with two possible outcomes, the total number of outcomes is \(2^3\).

b) i) To find the probability that all complete the race, multiply the probabilities of each completing the race: \(0.9 \times 0.7 \times 0.6\).

ii) To find the probability that at most one completes the race, consider the scenarios where none or only one completes the race. Calculate the probability for each scenario and sum them.

iii) To find the probability that at least two complete the race, consider the complementary probability of fewer than two completing the race and subtract it from 1.

The total number of outcomes for the three participants, Rickey, Ashley, and Loretta, is calculated as follows: \[ \text{Total Outcomes} = 2^3 = 8 \]

The probability that all three participants complete the race is given by the product of their individual probabilities: \[ P(\text{All Complete}) = P(Rickey) \times P(Ashley) \times P(Loretta) = 0.9 \times 0.7 \times 0.6 = 0.378 \]

To find the probability that at most one participant completes the race, we consider the following scenarios:

1. None complete the race: \[ P(\text{None Complete}) = (1 - 0.9) \times (1 - 0.7) \times (1 - 0.6) = 0.012 \]
2. Only Rickey completes the race: \[ P(\text{Only Rickey}) = 0.9 \times (1 - 0.7) \times (1 - 0.6) = 0.108 \]
3. Only Ashley completes the race: \[ P(\text{Only Ashley}) = (1 - 0.9) \times 0.7 \times (1 - 0.6) = 0.028 \]
4. Only Loretta completes the race: \[ P(\text{Only Loretta}) = (1 - 0.9) \times (1 - 0.7) \times 0.6 = 0.018 \]

Summing these probabilities gives: \[ P(\text{At Most One}) = P(\text{None Complete}) + P(\text{Only Rickey}) + P(\text{Only Ashley}) + P(\text{Only Loretta}) = 0.012 + 0.108 + 0.028 + 0.018 = 0.166 \]

The probability that at least two participants complete the race is the complement of the probability that at most one completes: \[ P(\text{At Least Two}) = 1 - P(\text{At Most One}) = 1 - 0.166 = 0.834 \]

Final Answer