Final answer:
To determine the joint probability distribution for X and Y, all possible combinations of apples and bananas in a sample of 4 fruits must be calculated. The probabilities of picking a particular jelly bean color are found by ratio of specific jelly beans to total count. Conditional probability such as P(A|B) for dice rolls is the ratio of favorable outcomes in A that also occur in B.
Step-by-step explanation:
Answering the given student's question involves a few steps of probability calculations and understanding the concepts of joint probability distributions and conditional probability. Working on the questions related to finding the probability of getting a specific type of jelly bean within a jar or calculating the probability of an event given another event has occurred is fundamental to grasping these concepts.
Joint Probability Distribution
To find the joint probability distribution of X (number of apples) and Y (number of bananas) when 4 fruits are chosen, you would list all possible combinations of apples (X) and bananas (Y) that could be chosen in the sample of 4. For each combination, you would calculate the probability using combinations and total possible outcomes.
Calculating Specific Probabilities
To calculate probabilities like P(B), P(G), or P(O), where B is the event of getting a blue jelly bean, G for a green, and O for an orange, you would divide the number of each type of jelly bean by the total number of jelly beans. For instance, to find P(B), divide the number of blue jelly beans by the total.
Conditional Probability
For conditional probability, such as P(A|B), where A and B are events with A = {2, 3} and B = {2, 4, 6} on a dice roll, the probability would be the probability that the outcome is in A given it is also in B. Since there's only one outcome in both A and B, the probability P(A|B) is 1/3, as there is only one favorable outcome over three possible outcomes within B.