Final answer:
To find the probabilities of various voting outcomes among 20 voters given a 70% chance of any individual voting, we use the binomial probability formula, calculating the probability for each specific scenario.
Step-by-step explanation:
We can tackle this problem by applying the binomial probability formula, which is used when we have a fixed number of independent trials, each with two possible outcomes (success or failure), and the probability of success is the same in each trial. In this case, 'success' refers to an eligible voter voting in the presidential election, with a probability of success given as p=0.70.
Part A
The probability that exactly 15 voters will vote is given by the formula:
P(X = k) = C(n, k) × p^k × (1-p)^(n-k)
Where:
- C(n, k) is the binomial coefficient, representing the number of ways to choose k successes from n trials.
- 'p' is the probability of success,
- 'k' is the number of successes,
- 'n' is the total number of trials.
For n=20 and k=15, the formula gives us P(X = 15).
Part B
The probability that at least 18 voters will vote is the sum of the probabilities of having 18, 19, or 20 voters vote. We calculate each separately and add them up:
P(X ≥ 18) = P(X = 18) + P(X = 19) + P(X = 20)
Part C
The probability that no more than 10 voters will vote is the sum of the probabilities of having 0 to 10 voters vote:
P(X ≤ 10) = P(X = 0) + P(X = 1) + … + P(X = 10)
Part D
The probability that more than 5 voters will vote is 1 minus the probability of having 5 or fewer voters vote (the complement of 'more than 5 voters will vote'):
P(X > 5) = 1 - P(X ≤ 5)
We apply the binomial formula to each scenario to find the respective probabilities.