Final answer:
The probability of selecting at least three point guards from five different basketball teams can be calculated by adding the probabilities of selecting exactly three, four, and five point guards. By using the formula for combinations and the probabilities of choosing point guards or non-point guards, we find that the probability is 0.076.
Step-by-step explanation:
To find the probability of selecting at least three point guards, we need to consider the number of ways this can occur.
First, we calculate the probability of selecting exactly three point guards. There are 5 choose 3 ways to select 3 point guards from the 5 teams, which is equal to 10.
Each of these 10 combinations has a (1/5) * (1/5) * (1/5) * (4/5) * (4/5) probability of occurring since we are choosing 3 point guards and 2 non-point guard positions.
Next, we calculate the probability of selecting exactly four point guards. There are 5 choose 4 ways to select 4 point guards from the 5 teams, which is equal to 5.
Each of these 5 combinations has a (1/5) * (1/5) * (1/5) * (1/5) * (4/5) probability of occurring.
Finally, we calculate the probability of selecting exactly five point guards. There is only 1 way to select 5 point guards from the 5 teams.
This combination has a (1/5) * (1/5) * (1/5) * (1/5) * (1/5) probability of occurring.
Therefore, the total probability of selecting at least three point guards is the sum of these probabilities: (10 * ((1/5)^3) * ((4/5)^2)) + (5 * ((1/5)^4) * ((4/5)^1)) + (1 * ((1/5)^5)).
Simplifying this expression, we get a probability of 0.076 (option A).