*Give all probabilities to four decimal places.
Bowl #1 contains 4 grape candies, 5 lemon candies, 6 cherry candies and 5 raspberry candies.
Bowl #2 contains 8 grape candies, 5 lemon candies, 4 cherry candies and 3 raspberry candies.
(c) What is the probability that the two selected candies are the same flavour?
(d) What is the probability that the two selected candies are different colours?
(e) What is the probability that the first selected candy is lemon or that the second selected candy is cherry?
(f) Let X be the number of grape candies that are selected. Find the probability distribution of X by filling in the table below:
X |0| |1| |2|
P(X = x)| | | | | |
(c) To calculate the probability that the two selected candies are the same flavor, we need to consider the total number of candies in both bowls and the number of candies of each flavor in each bowl.
First, let's calculate the probability of selecting two candies of the same flavor from Bowl #1. There are a total of 20 candies in Bowl #1, so the probability of selecting two candies of the same flavor from Bowl #1 would be:
Probability from Bowl #1 = (Probability of selecting grape candies from Bowl #1) + (Probability of selecting lemon candies from Bowl #1) + (Probability of selecting cherry candies from Bowl #1) + (Probability of selecting raspberry candies from Bowl #1)
= (C(4, 2) / C(20, 2)) + (C(5, 2) / C(20, 2)) + (C(6, 2) / C(20, 2)) + (C(5, 2) / C(20, 2))
Where C(n, k) represents the number of combinations of k items from a set of n items.
Similarly, let's calculate the probability of selecting two candies of the same flavor from Bowl #2:
Probability from Bowl #2 = (Probability of selecting grape candies from Bowl #2) + (Probability of selecting lemon candies from Bowl #2) + (Probability of selecting cherry candies from Bowl #2) + (Probability of selecting raspberry candies from Bowl #2)
= (C(8, 2) / C(20, 2)) + (C(5, 2) / C(20, 2)) + (C(4, 2) / C(20, 2)) + (C(3, 2) / C(20, 2))
Now, to find the overall probability that the two selected candies are the same flavor, we need to add the probabilities from both bowls:
Probability (c) = Probability from Bowl #1 + Probability from Bowl #2
(d) To calculate the probability that the two selected candies are different colors, we can use similar approach as in (c). We need to calculate the probability of selecting a candy of each flavor from Bowl #1 and then multiply it by the probability of selecting a candy of a different flavor from Bowl #2. Then, we add the probabilities for each flavor to get the overall probability.
Probability (d) = (Probability of selecting grape candies from Bowl #1) * (Probability of selecting non-grape candy from Bowl #2) + (Probability of selecting lemon candies from Bowl #1) * (Probability of selecting non-lemon candy from Bowl #2) + (Probability of selecting cherry candies from Bowl #1) * (Probability of selecting non-cherry candy from Bowl #2) + (Probability of selecting raspberry candies from Bowl #1) * (Probability of selecting non-raspberry candy from Bowl #2)
(e) To calculate the probability that the first selected candy is lemon or the second selected candy is cherry, we need to calculate the probability of selecting a lemon candy as the first candy, and then multiply it by the probability of selecting any candy as the second candy. Similarly, we need to calculate the probability of selecting any candy as the first candy, and then multiply it by the probability of selecting a cherry candy as the second candy. Finally, we add these probabilities to get the overall probability.
Probability (e) = (Probability of selecting lemon candy as first candy) * (Probability of selecting any candy as second candy) + (Probability of selecting any candy as first candy) * (Probability of selecting cherry candy as second candy)
(f) To find the probability distribution of X, which represents the number of grape candies selected, we need to calculate the probability of selecting 0, 1, or 2 grape candies.
P(X = 0) = (Probability of not selecting any grape candies from Bowl #1) * (Probability of not selecting any grape candies from Bowl #2)
P(X = 1) = (Probability of selecting 1 grape candy from Bowl #1) * (Probability of not selecting any grape candies from Bowl #2) + (Probability of not selecting any grape candies from Bowl #1) * (Probability of selecting 1 grape candy from Bowl #2)
P(X = 2) = (Probability of selecting 2 grape candies from Bowl #1) * (Probability of not selecting any grape candies from Bowl #2)
Note: The probabilities for selecting grape candies can be calculated in a similar manner as shown in (c) and (d), replacing other flavors with grape candies. The probabilities for not selecting grape candies can be calculated by subtracting the probability of selecting grape candies from 1.