Final answer:
The joint probability distribution of X_1 and Y_1, representing the number of heads tossed by two children playing a coin-toss game, is found by considering all possible outcomes of tossing two fair coins and multiplying the individual probabilities since the events are independent.
Step-by-step explanation:
To find the joint probability distribution of X1 and Y1, we must consider the possible outcomes for each child (A and J) when they toss two fair coins. Since each coin is fair, the probability of getting heads (H) or tails (T) is 0.5.
Each child can get 0, 1, or 2 heads. The sample space for each child's two-coin toss is {HH, HT, TH, TT}, and the probabilities are:
- P(2 heads) = P(HH) = (0.5)(0.5) = 0.25
- P(1 head) = P(HT or TH) = (0.5)(0.5) + (0.5)(0.5) = 0.5
- P(0 heads) = P(TT) = (0.5)(0.5) = 0.25
Therefore, the possible outcomes for X1 and Y1 are (0, 0), (0, 1), (0, 2), (1, 0), (1, 1), (1, 2), (2, 0), (2, 1), and (2, 2), with both children's results independent of each other. The joint probabilities can be found by multiplying the probabilities of the individual outcomes. For example:
- P(X1=0, Y1=0) = P(A gets TT) × P(J gets TT) = 0.25 × 0.25 = 0.0625
- P(X1=1, Y1=1) = P(A gets HT or TH) × P(J gets HT or TH) = 0.5 × 0.5 = 0.25
Complete the process to find all joint probabilities, and you will have the full joint probability distribution for X1 and Y1.