Here is information on the results of the 90 members of an athletics team at a competition.
If an athlete is chosen at random, the probability that he or she won!
- a gold medal is 1/6
- a silver medal is 0.3
- a bronze medal is 0.4
• Four athletes won gold and silver medals.
• Six athletes won bronze and silver medals.
• Seven athletes won gold and bronze medals.
• Two athletes won all three types of medals.
If an athlete from this team is randonly selected, what is the probability that he or she:
a) won a silver or bronze medal Knowing that he or she also won a gold medal?
b) won a gold and bronze medal knowing that he or she also won a silver medal?
c) has won atleast one other type of medal, Knowing that he or she has also won a bronze medal?
To answer these probability questions, we need to use conditional probability. Conditional probability is the probability of an event (in this case, winning a certain medal) occurring given that another event (in this case, winning a different medal) has already occurred.
To calculate conditional probability, we use the formula:
P(A|B) = P(A and B) / P(B)
Where P(A|B) is the probability of event A occurring given that event B has already occurred, P(A and B) is the probability of both events A and B occurring, and P(B) is the probability of event B occurring.
Now let's solve the given probability questions:
a) To find the probability that an athlete won a silver or bronze medal, given that they also won a gold medal, we want to calculate P(Silver or Bronze|Gold).
We know that the probability of winning a gold medal is 1/6. We also know that there are four athletes who won both gold and silver medals, and seven athletes who won both gold and bronze medals. However, we need to consider that the two athletes who won all three types of medals have been counted twice in these numbers, so we need to subtract them once.
Therefore, the number of athletes who won a gold medal is (4 + 7 - 2) = 9.
The probability of winning a silver or bronze medal, given that they won a gold medal, is the number of athletes who won both a gold and silver/bronze medal divided by the total number of athletes who won a gold medal. So:
P(Silver or Bronze|Gold) = (4 + 6 - 2) / 9 = 8 / 9
Therefore, the probability that an athlete won a silver or bronze medal, given that they also won a gold medal, is 8/9.
b) To find the probability that an athlete won a gold and bronze medal, given that they also won a silver medal, we want to calculate P(Gold and Bronze|Silver).
We know that the probability of winning a silver medal is 0.3. We also know that there are four athletes who won both gold and silver medals, and six athletes who won both bronze and silver medals. However, we need to consider that the two athletes who won all three types of medals have been counted twice in these numbers, so we need to subtract them once.
Therefore, the number of athletes who won a silver medal is (4 + 6 - 2) = 8.
The probability of winning a gold and bronze medal, given that they won a silver medal, is the number of athletes who won both a gold and bronze medal and also won a silver medal, divided by the total number of athletes who won a silver medal. So:
P(Gold and Bronze|Silver) = 2 / 8 = 1 / 4
Therefore, the probability that an athlete won a gold and bronze medal, given that they also won a silver medal, is 1/4.
c) To find the probability that an athlete has won at least one other type of medal, given that they have also won a bronze medal, we want to calculate P(At least one other medal|Bronze).
We know that the probability of winning a bronze medal is 0.4. We also know that there are seven athletes who won both gold and bronze medals, and six athletes who won both bronze and silver medals. However, we need to consider that the two athletes who won all three types of medals have been counted twice in these numbers, so we need to subtract them once.
Therefore, the number of athletes who won a bronze medal is (7 + 6 - 2) = 11.
The probability of winning at least one other medal, given that they won a bronze medal, is the number of athletes who won either a gold or silver medal (excluding those who won all three types of medals), divided by the total number of athletes who won a bronze medal. So:
P(At least one other medal|Bronze) = (4 + 6 - 2) / 11 = 8 / 11
Therefore, the probability that an athlete has won at least one other type of medal, given that they have also won a bronze medal, is 8/11.