The chances of three independent events P, Q, R occuring are 1/2, 2/3, 1/4 respectively. What are the chances of two of them only occuring
To find the chances of only two events occurring, we need to consider all possible pairs of events and add their individual probabilities.
For P and Q to occur:
Probability = (1/2) x (2/3) x (3/4)
= 1/4
For P and R to occur:
Probability = (1/2) x (1/3) x (3/4)
= 1/8
For Q and R to occur:
Probability = (1/2) x (2/3) x (1/4)
= 1/12
So the total probability of only two events occurring is:
1/4 + 1/8 + 1/12
= 5/12
To find the chances of two of the events occurring, we need to consider all possible combinations of two events out of the three.
There are three possible combinations where two events can occur:
1. P and Q
2. P and R
3. Q and R
For each combination, we need to multiply the probabilities of the two events occurring and the probability of the remaining event not occurring.
Let's calculate the chances for each combination:
1. P and Q:
Probability of P and Q occurring: (1/2) * (2/3) = 1/3
Probability of R not occurring: 1 - 1/4 = 3/4
Probability of two events occurring and one not occurring: (1/3) * (3/4) = 1/4
2. P and R:
Probability of P and R occurring: (1/2) * (1/4) = 1/8
Probability of Q not occurring: 1 - 2/3 = 1/3
Probability of two events occurring and one not occurring: (1/8) * (1/3) = 1/24
3. Q and R:
Probability of Q and R occurring: (2/3) * (1/4) = 1/6
Probability of P not occurring: 1 - 1/2 = 1/2
Probability of two events occurring and one not occurring: (1/6) * (1/2) = 1/12
Now, we add up the probabilities for each combination to get the overall chance of two events only occurring:
(1/4) + (1/24) + (1/12) = 3/24 + 1/24 + 2/24 = 6/24 = 1/4
Therefore, the chances of two of the three events occurring is 1/4.