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.
To find the probability that a person owns either a microwave or a CD player, but not both, we need to find the sum of the probabilities of owning only a microwave and owning only a CD player.
Let's define:
A = event that a person owns a microwave
B = event that a person owns a CD player
We are given:
P(A) = 0.75 (probability of owning a microwave)
P(B) = 0.25 (probability of owning a CD player)
P(A ∩ B) = 0.16 (probability of owning both a microwave and a CD player)
Now, let's find the probability of owning only a microwave:
P(A only) = P(A) - P(A ∩ B)
= 0.75 - 0.16
= 0.59
Similarly, let's find the probability of owning only a CD player:
P(B only) = P(B) - P(A ∩ B)
= 0.25 - 0.16
= 0.09
Now, to find the probability that a person owns either a microwave or a CD player, but not both, we need to find the sum of P(A only) and P(B only):
P(A only or B only) = P(A only) + P(B only)
= 0.59 + 0.09
= 0.68
Therefore, the probability that a person owns either a microwave or a CD player, but not both, is 0.68.
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:
P(A) = 0.75 (probability that a person owns a microwave oven)
P(B) = 0.25 (probability that a person owns a compact disk player)
P(A ∩ B) = 0.16 (probability that a person owns both a microwave and a CD player)
We want to find the probability that a person owns either a microwave or a CD player, but not both.
To calculate this probability, we need to find the union of events A and B (A ∪ B), excluding the intersection of events A and B (A ∩ B).
P(A ∪ B, not A ∩ B) = P(A) + P(B) - 2P(A ∩ B)
Substituting the given values:
P(A ∪ B, not A ∩ B) = 0.75 + 0.25 - 2 * 0.16
P(A ∪ B, not A ∩ B) = 0.84
Therefore, the probability that a person owns either a microwave or a CD player, but not both, is 0.84.