In 2011 a spots club offers its members three types of activities: volleyball basketball and tennis.
Each member practices only one of the three activities:30% of the members practice volleyball, 20% of the members practice basketball, the rest of the members practice tennis.
The club suggests a reunion day for all its members: 20% of the members who practice volleyball participate in the reunion, 25% of the members who practice basketball participate in the reunion, 70% of the members who practice tennis participate in the reunion.
We randomly choose one member of the club. Consider the following events:
V:The chosen member practices volleyball
B:The chosen member practices basketball
T:The chosen member practices tennis
R:The chosen member attends the reunion
1)Show that the probability P(TUR) is equal to o.35, calculate P(B intersection R) and P(V intersection R).
2) The president of the club claims that half of the members are not participating in the reunion.
Justify his claim by calculations.
THANKS IN ADVANCE
To calculate the probabilities, we need to use the given information about the members who practice each sport and the participants in the reunion.
1) Probability of TUR (Tennis, Union, Reunion):
We know that 30% of the members practice volleyball (V), 20% practice basketball (B), and the rest practice tennis (T). So the percentage of members who practice tennis can be calculated as 100% - (30% + 20%) = 50%. Since 70% of the members who practice tennis participate in the reunion, we can find the probability of TUR as follows:
P(TUR) = P(T) * P(R|T) = 0.5 * 0.7 = 0.35
Probability of B intersection R (Basketball, Intersection, Reunion):
We already know that 20% of the members practice basketball (B), and 25% of them participate in the reunion (R). To find the probability of B intersection R, we can calculate the intersection of these two events:
P(B intersection R) = P(B) * P(R|B) = 0.2 * 0.25 = 0.05
Probability of V intersection R (Volleyball, Intersection, Reunion):
Given that 30% of the members practice volleyball (V) and 20% of them participate in the reunion (R), we can calculate the intersection of these two events:
P(V intersection R) = P(V) * P(R|V) = 0.3 * 0.2 = 0.06
2) To justify the president's claim that half of the members are not participating in the reunion, we need to compare the probability of not participating (complement) to the probability of participating.
Probability of not participating in the reunion (not R):
Since the reunion includes members who practice volleyball, basketball, and tennis, we can find the probability of not participating (not R) by subtracting the probabilities of participating from 1:
P(not R) = 1 - (P(R|V) * P(V) + P(R|B) * P(B) + P(R|T) * P(T))
= 1 - (0.2 * 0.3 + 0.25 * 0.2 + 0.7 * 0.5)
= 1 - (0.06 + 0.05 + 0.35)
= 1 - 0.46
= 0.54
Now we can compare the probability of not participating (0.54) to the probability of participating (0.46). Since the probability of not participating is greater, we can conclude that more than half of the members are not participating in the reunion, supporting the president's claim.
Therefore, the president's claim that half of the members are not participating in the reunion is justified by calculations.