Of all the Sunny Club members in a particular city, 25% prefer swimming on weekends and 75% prefer swimming on weekdays. 10% of the members in the city prefer swimming on weekends and are female. 55% of the members in the city prefer swimming on weekdays and are female. What is the probability that a randomly selected club member is female, given that the person prefers swimming on weekends?
Contestants in a game show are each given three pairs of numbers (for example, 2-3, 5-6, and 6-1). A person in the audience is randomly selected to roll two dice, one white and one red, where white symbolizes the first number in the pair and red the second number. What is the probability that each player has of winning?
To find the probability that a randomly selected club member is female, given that the person prefers swimming on weekends, we can use Bayes' theorem.
Let's define the following events:
A: The person is female
B: The person prefers swimming on weekends
We are given the following information:
P(A) = 10% (10% of the members prefer swimming on weekends and are female)
P(B) = 25% (25% of the members prefer swimming on weekends)
P(A|B) = ? (the probability that a randomly selected club member is female, given that the person prefers swimming on weekends)
According to Bayes' theorem, we have:
P(A|B) = (P(B|A) * P(A)) / P(B)
Now, let's find the missing probabilities:
P(B|A) is the probability that a member prefers swimming on weekends, given that the person is female. From the given information, it is not explicitly mentioned. We can assume that it is the same as P(A|B) since both percentages are about the same group of people who prefer swimming on weekends and are female.
Therefore, P(B|A) = P(A|B)
Next, we find P(B) using the Law of Total Probability:
P(B) = P(B|A) * P(A) + P(B|not A) * P(not A)
P(B) = P(A|B) * P(A) + P(not A|B) * P(not A)
Since the information about P(B|not A) and P(not A|B) is not given, we cannot directly calculate P(B). However, we can utilize the fact that the total probability of all club members who prefer swimming on weekdays is 75% and 55% of them are female.
P(prefer swimming on weekdays) = 75%
P(female|prefer swimming on weekdays) = 55%
We can assume that the remaining percentage, 45% (100% - 55%), refers to the male members who prefer swimming on weekdays.
From this information, we can calculate P(not A) as follows:
P(not A) = P(prefer swimming on weekdays) - P(female|prefer swimming on weekdays)
P(not A) = 75% - 55%
P(not A) = 20%
Now, let's substitute all these values into Bayes' theorem:
P(A|B) = (P(B|A) * P(A)) / P(B)
= (P(A|B) * 10%) / P(B)
P(B) = P(A|B) * P(A) + P(not A|B) * P(not A)
= P(A|B) * 10% + P(not A|B) * 20%
Since P(B) = 25% (the probability of preferring swimming on weekends), we have:
25% = (P(A|B) * 10%) / (P(A|B) * 10% + P(not A|B) * 20%)
Now, solving this equation will give us the value of P(A|B), which is the probability that a randomly selected club member is female, given that the person prefers swimming on weekends.
To find the probability that a randomly selected club member is female, given that the person prefers swimming on weekends, we need to use conditional probability.
Let's break down the information given:
- Of all the Sunny Club members in the city, 25% prefer swimming on weekends. This means that 25% of the total members (regardless of gender) prefer swimming on weekends.
- Of the members who prefer swimming on weekends, 10% are female. This means that 10% of the members who prefer swimming on weekends are female.
- Of all the Sunny Club members in the city, 75% prefer swimming on weekdays. This means that 75% of the total members (regardless of gender) prefer swimming on weekdays.
- Of the members who prefer swimming on weekdays, 55% are female. This means that 55% of the members who prefer swimming on weekdays are female.
Now let's calculate the probability using the formula for conditional probability:
P(Female | Weekends) = P(Female and Weekends) / P(Weekends)
To find P(Female and Weekends), we multiply the probability of being female given weekends (10%) by the probability of preferring weekends (25%):
P(Female and Weekends) = (10% of all members) * (25% of all members) = 0.1 * 0.25 = 0.025
Next, we need to calculate P(Weekends), which is the probability of preferring weekends:
P(Weekends) = 25% of all members
Finally, we can substitute these values in the formula for conditional probability:
P(Female | Weekends) = P(Female and Weekends) / P(Weekends) = 0.025 / 0.25 = 0.1
Therefore, the probability that a randomly selected club member is female, given that the person prefers swimming on weekends, is 0.1 or 10%.