a. To calculate the probability of getting more than 65% green balls, we need to find the probability of getting 0.65 or less green balls, and subtract it from 1.
First, let's calculate the probability of getting 0.65 or less green balls.
The total number of balls in the machine is 93 + 65 = 158.
For 90 draws, the expected number of green balls is 0.65 * 90 = 58.5
We can use the binomial probability formula to find the probability of getting 0.65 or less green balls:
P(X ≤ 58) = ∑ [90 C x * (0.65)^x * (0.35)^(90-x)], where x ranges from 0 to 58.
Using the z-table or software, we can find that P(X ≤ 58) ≈ 0.0390
Now, we can calculate the probability of getting more than 65% green balls:
P(X > 58) = 1 - P(X ≤ 58) ≈ 1 - 0.0390 = 0.9610
So, the probability of getting more than 65% green balls is approximately 0.9610.
b. To determine whether to choose 90 or 120 draws, we need to compare the probabilities of getting more than 65% green balls for each scenario.
For 120 draws, the expected number of green balls is 0.65 * 120 = 78
We need to calculate the probability of getting more than 65% green balls for 120 draws.
Using the same binomial probability formula as above, we can find:
P(X > 78) = 1 - P(X ≤ 78)
Using the z-table or software, we can find that P(X ≤ 78) ≈ 0.9309
P(X > 78) = 1 - 0.9309 = 0.0691
Therefore, the probability of getting more than 65% green balls for 120 draws is approximately 0.0691.
Comparing the probabilities, we can see that the probability of getting more than 65% green balls is higher for 90 draws (0.9610) than for 120 draws (0.0691).
Therefore, I would choose 90 draws for the game.
At an exhibit in the Museum of Science, people are asked to choose between 90 or 120 random draws from a machine. The machine is known to have 93 green balls and 65 red balls. After each draw, the color of the ball is noted and the ball is put back for the next draw. You win a prize if more than 65% of the draws result in a green ball. [You may find it useful to reference the z table.]
a. Calculate the probability of getting more than 65% green balls. (Round final answer to 4 decimal places.)
b. Would you choose 90 or 120 draws for the game?
multiple choice
90 balls
120 balls
1 answer