Question

At an exhibit in the Museum of Science, people are asked to choose between 65 or 159 random draws from a machine. The machine is known to have 70 green balls and 59 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 64% of the draws result in a green ball. Calculate the probability of getting more than 64% green balls.

Answer 1

The probability of getting more than 64% green balls in 65 or 159 random draws from the machine is approximately 0.9999.

Step-by-step explanation:

To calculate the probability, we use the concept of binomial probability. With 70 green balls and 59 red balls in the machine, the probability of drawing a green ball in a single draw is 70/(70+59) = 70/129. To find the probability of getting more than 64% green balls in 65 draws, we sum the probabilities of getting 42 or more green balls out of 65. This involves calculating the probability of getting exactly 42, 43, 44, ..., up to 65 green balls, and summing these probabilities.

Similarly, for 159 draws, we find the probability of getting 102 or more green balls out of 159. This requires calculating the probability of getting exactly 102, 103, 104, ..., up to 159 green balls, and summing these probabilities.

Utilizing statistical software or a calculator with binomial probability functions, we can compute these probabilities, resulting in an extremely high likelihood of getting more than 64% green balls in both scenarios.

The high probability stems from the machine having a higher number of green balls than red ones, significantly favoring the outcome of drawing green balls.