What is the probability that after 3 hours of randomly picking balls every minute from a box containing one yellow ball and one green ball (starting from just these two balls), the box will contain exactly 85 green balls?
To find the probability of obtaining exactly 85 green balls after 3 hours of randomly picking balls every minute, we need to determine the number of possible outcomes that satisfy this condition and divide it by the total number of possible outcomes.
Let's break down the problem step by step:
1. Each minute, you randomly pick one ball from the box. Since the box initially contains one yellow ball and one green ball, the probability of picking a green ball in any given minute is 1/2, and the probability of picking a yellow ball is also 1/2.
2. After 3 hours, you would have picked 3 hours × 60 minutes/hour = 180 balls from the box.
3. To have exactly 85 green balls after picking 180 balls, you need to calculate the number of ways to choose (180 - 85) = 95 yellow balls out of the remaining 181 balls (since you only have one yellow ball left in the box at that point).
4. The total number of possible outcomes is obtained by calculating the total number of ways to choose 180 balls from the remaining 181 balls in the box.
5. Finally, calculate the probability by dividing the number of favorable outcomes (choosing 95 yellow balls out of 181) by the total number of possible outcomes (choosing 180 balls out of 181).
Now you can use this information to calculate the probability using combinatorial mathematics or programming techniques like binomial coefficients.