A fair coin is tossed, and a fair die is thrown. Write down sample spaces for(a) the toss of the coin;(b) the throw of the die;(c) the combination of these experiments.Let A be the event that a head is tossed, and B be the event that an odd number isthrown. Directly from the sample space, calculate P (A ∪ B) and P (A ∩ B)
Question
A fair coin is tossed, and a fair die is thrown. Write down sample spaces for(a) the toss of the coin;(b) the throw of the die;(c) the combination of these experiments.Let A be the event that a head is tossed, and B be the event that an odd number isthrown. Directly from the sample space, calculate P (A ∪ B) and P (A ∩ B)
Solution
(a) The sample space for the toss of the coin is {H, T} where H represents heads and T represents tails.
(b) The sample space for the throw of the die is {1, 2, 3, 4, 5, 6}.
(c) The combination of these experiments would be {(H, 1), (H, 2), (H, 3), (H, 4), (H, 5), (H, 6), (T, 1), (T, 2), (T, 3), (T, 4), (T, 5), (T, 6)}.
Let A be the event that a head is tossed, and B be the event that an odd number is thrown.
The event A = {(H, 1), (H, 2), (H, 3), (H, 4), (H, 5), (H, 6)} and the event B = {(H, 1), (H, 3), (H, 5), (T, 1), (T, 3), (T, 5)}.
P(A ∪ B) is the probability of either event A or event B occurring. This is equal to the number of outcomes in A ∪ B divided by the total number of outcomes. A ∪ B = {(H, 1), (H, 2), (H, 3), (H, 4), (H, 5), (H, 6), (T, 1), (T, 3), (T, 5)}. There are 9 outcomes in A ∪ B and 12 total outcomes, so P(A ∪ B) = 9/12 = 0.75.
P(A ∩ B) is the probability of both event A and event B occurring. This is equal to the number of outcomes in A ∩ B divided by the total number of outcomes. A ∩ B = {(H, 1), (H, 3), (H, 5)}. There are 3 outcomes in A ∩ B and 12 total outcomes, so P(A ∩ B) = 3/12 = 0.25.
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