The probability that a person owns a microwave oven is .75, that a person owns a compact disk player is .25, and that a person owns both a microwave and a CD player is .16. Find the probability that a person owns either a microwave or a CD player, but not both.
p(not having both)=0.84
=0.84-0.16
=0.68
To find the probability that a person owns either a microwave or a CD player, but not both, we need to subtract the probability that a person owns both from the sum of the probabilities that a person owns a microwave and a CD player separately.
Let's call the probability that a person owns a microwave M and the probability that a person owns a CD player C.
Given:
P(M) = 0.75 (probability a person owns a microwave)
P(C) = 0.25 (probability a person owns a CD player)
P(M∩C) = 0.16 (probability a person owns both a microwave and a CD player)
To find the probability that a person owns either a microwave or a CD player, but not both, we can use the formula:
P(A∪B) = P(A) + P(B) - 2(P(A∩B))
In this case, A represents owning a microwave, and B represents owning a CD player.
Substituting the given values into the formula:
P(A∪B) = P(M) + P(C) - 2(P(M∩C))
P(A∪B) = 0.75 + 0.25 - 2(0.16)
P(A∪B) = 0.75 + 0.25 - 0.32
P(A∪B) = 0.93
Therefore, the probability that a person owns either a microwave or a CD player, but not both, is 0.93 or 93%.
To find the probability that a person owns either a microwave or a CD player, but not both, we can use the principle of inclusion-exclusion.
Let's define the following events:
A: A person owns a microwave oven
B: A person owns a compact disk player
We are given the following probabilities:
P(A) = 0.75 (Probability that a person owns a microwave oven)
P(B) = 0.25 (Probability that a person owns a CD player)
P(A ∩ B) = 0.16 (Probability that a person owns both a microwave and a CD player)
The probability that a person owns either a microwave or a CD player, but not both, is given by the expression P((A ∪ B) - (A ∩ B)).
To calculate this, we need to calculate P(A ∪ B) and P(A ∩ B).
Using the principle of inclusion-exclusion, we have:
P(A ∪ B) = P(A) + P(B) - P(A ∩ B)
Plugging in the given probabilities:
P(A ∪ B) = 0.75 + 0.25 - 0.16
P(A ∪ B) = 0.84
Now, we need to calculate P(A ∩ B):
P(A ∩ B) = 0.16 (Given)
Finally, we can calculate the probability that a person owns either a microwave or a CD player, but not both:
P((A ∪ B) - (A ∩ B)) = P(A ∪ B) - P(A ∩ B)
P((A ∪ B) - (A ∩ B)) = 0.84 - 0.16
P((A ∪ B) - (A ∩ B)) = 0.68
Therefore, the probability that a person owns either a microwave or a CD player, but not both, is 0.68.