A bag holds 20 yellow mints and 80 other green or pink mints. You choose a mint at random, eat it, and choose another.
Find the number of pink mints if P(yellow then pink) = P(green then yellow).
What is the least number of pink mints if P(yellow then pink) = P(green then yellow)?
To find the number of pink mints if P(yellow then pink) = P(green then yellow), we can start by determining the probabilities of each sequence of mint choices.
Let's denote the number of pink mints as "P", and the total number of mints as "T". From the information given in the question, we know that there are 20 yellow mints and 80 other mint(s), which can be either green or pink. Therefore, we can write the equation:
T = 20 (yellow) + 80 (green or pink) = 100
Now, let's consider the probability of choosing a yellow mint and then a pink mint (P(yellow then pink)). The probability of choosing a yellow mint first is 20/100 because there are 20 yellow mints out of a total of 100 mints.
After choosing a yellow mint and eating it, the total number of mints remaining is now 99 (one mint has been removed from the bag). out of these remaining mints, the number of pink mints is P since we want to determine the minimum number of pink mints satisfying the given condition.
Therefore, the probability of choosing a pink mint from the remaining mints after choosing a yellow mint can be expressed as P/99.
For P(green then yellow), we need to consider the probability of choosing a green mint first. Since there are 80 mints that are either green or pink, and P pink mints remaining, the number of green mints must be 80 - P (green + pink = 80).
The probability of choosing a green mint first is (80 - P)/100 because there are (80 - P) green mints out of a total of 100 mints.
Following the same logic as before, after choosing a green mint and eating it, the probability of choosing a yellow mint from the remaining mints is 20/(99 - 1) = 20/98.
Now, let's set up the equation and solve for P:
P(yellow then pink) = P(green then yellow)
(20/100) * (P/99) = ((80 - P)/100) * (20/98)
Simplifying the equation:
20P = (80 - P) * 99
20P = 7920 - 99P
119P = 7920
P ≈ 66
Therefore, the least number of pink mints is approximately 66, if P(yellow then pink) = P(green then yellow).