in a group of 100 people, 40 own a cat, 25 own a dog and 15 own a cat and a dog. Find the probability that a person chosen at random
a) owns a dog or a cat
b) owns a dog or a cat, but not both
c)owns a dog, given that he owns a cat
d) does not own a cat, given that he owns a dog.
A: 80/100
B: 65/100
C: 15/100
D: 25/100
To solve these probability problems, we need to start by analyzing the given information.
Let's define:
A = the event that a person owns a cat
B = the event that a person owns a dog
According to the information provided:
P(A) = 40 (out of 100 own a cat)
P(B) = 25 (out of 100 own a dog)
P(A ∩ B) = 15 (out of 100 own a cat and a dog)
Now we can proceed to answer the specific questions:
a) To find the probability that a person chosen at random owns a dog or a cat (P(A ∪ B)), we can use the formula for the union of two events:
P(A ∪ B) = P(A) + P(B) - P(A ∩ B)
Substituting the values:
P(A ∪ B) = 40/100 + 25/100 - 15/100
P(A ∪ B) = 50/100
P(A ∪ B) = 0.5
So the probability that a person chosen at random owns a dog or a cat is 0.5 or 50%.
b) To find the probability that a person chosen at random owns a dog or a cat, but not both (P(A ∪ B) - P(A ∩ B)), we subtract the probability of owning both from the probability of owning either:
P(A ∪ B) - P(A ∩ B) = 0.5 - 15/100
P(A ∪ B) - P(A ∩ B) = 0.5 - 0.15
P(A ∪ B) - P(A ∩ B) = 0.35
Therefore, the probability that a person chosen at random owns a dog or a cat, but not both, is 0.35 or 35%.
c) To find the probability that a person chosen at random owns a dog, given that they own a cat (P(B|A)), we use the formula for conditional probability:
P(B|A) = P(A ∩ B) / P(A)
Substituting the values:
P(B|A) = 15/100 / 40/100
P(B|A) = 15/40
P(B|A) = 0.375
So the probability that a person chosen at random owns a dog, given that they own a cat, is 0.375 or 37.5%.
d) To find the probability that a person chosen at random does not own a cat, given that they own a dog (P(¬A|B)), we use the formula for conditional probability:
P(¬A|B) = P(¬A ∩ B) / P(B)
To calculate P(¬A ∩ B), we subtract the probability of owning both from the probability of owning a dog:
P(¬A ∩ B) = P(B) - P(A ∩ B)
P(¬A ∩ B) = 25/100 - 15/100
P(¬A ∩ B) = 10/100
Substituting the values:
P(¬A|B) = 10/100 / 25/100
P(¬A|B) = 10/25
P(¬A|B) = 0.4
Therefore, the probability that a person chosen at random does not own a cat, given that they own a dog, is 0.4 or 40%.