A sample space S yields six equally likely events, K, L, M, N, O, and P.
a. Find P(N). (Round your answer to 2 decimal places.)
b. Find P(Lc). (Round your answer to 2 decimal places.)
c. Find P(K U M U O). (Do not round intermediate calculations. Round your answer to 2 decimal places.)
a. Since the events in the sample space are equally likely, the probability of each event is 1/6.
Therefore, P(N) = 1/6.
b. The complement of event L, denoted by Lc, is the event that L does not occur.
Since each event in the sample space is equally likely, the probability of an event not occurring is the complement of the probability of the event occurring.
Therefore, P(Lc) = 1 - P(L) = 1 - 1/6 = 5/6.
c. The union of events K, M, and O (denoted as K U M U O) is the event that at least one of these three events occurs.
To find the probability of this union, we can sum the probabilities of each individual event and subtract the overlapping probabilities.
P(K U M U O) = P(K) + P(M) + P(O) - P(K ∩ M) - P(K ∩ O) - P(M ∩ O) + P(K ∩ M ∩ O).
Since each event is equally likely, this simplifies to:
P(K U M U O) = 1/6 + 1/6 + 1/6 - 0 - 0 - 0 + 0 = 3/6 = 1/2.
Therefore, P(K U M U O) = 1/2.
a. Since all the events in the sample space are equally likely, the probability of each event is given by P(event) = 1/total number of events. Therefore, P(N) = 1/6 ≈ 0.17.
b. The complement of an event is the event not occurring. Therefore, P(Lc) = 1 - P(L). Since all the events are equally likely, P(L) = 1/6. Thus, P(Lc) = 1 - 1/6 = 5/6 ≈ 0.83.
c. To find the probability of the union of multiple events, we need to add the probabilities of those events. Therefore, P(K U M U O) = P(K) + P(M) + P(O). Since all the events are equally likely, P(K) = P(M) = P(O) = 1/6. Thus, P(K U M U O) = 1/6 + 1/6 + 1/6 = 3/6 = 1/2 = 0.5.
To find the probability of an event, we need to divide the number of favorable outcomes by the total number of possible outcomes.
a. To find P(N), we need to know the number of favorable outcomes for N and the total number of possible outcomes. Since all events are equally likely, each event has the same probability. Therefore, the probability of each event is 1/6.
So, P(N) = 1/6.
b. To find P(Lc), we need to know the probability of the complement of event L, which means the probability of all events that are not L.
In this case, the complement of event L includes events K, M, N, O, and P, which are 5 out of the 6 possible events.
So, P(Lc) = 5/6.
c. To find P(K U M U O), we need to find the probability of the union of events K, M, and O.
The union of events means all outcomes that are in any of the events K, M, or O. In this case, the events are mutually exclusive, meaning no outcome can be in more than one of these events.
So, to get the probability of the union, we can simply add the probabilities of each event individually.
P(K U M U O) = P(K) + P(M) + P(O) = 1/6 + 1/6 + 1/6 = 3/6 = 1/2.
Therefore, P(K U M U O) = 1/2.