# Finding mutually exclusive events

A cricket club has $15$ members, of whom only $5$ can bowl. What is the probability that in a team of $11$ members at least $3$ bowlers are selected?

This is my textbook problem. Considering $A , B$ and $C$ to be three possible events, my book says they're mutually exclusive events. But how is that possible? I don't understand. Out of $15$ players, there cannot be three mutually exclusive team consisting of $11$ players. Somebody please help.

Let $A_k$ be the event that exactly $k$ bowlers are selected out of the group of $5$ bowlers for $k=0,1,\dots,5$. They are mutually exclusive events. Note that mutually exclusive does not mean that a team in $A_4$ and another team in $A_5$ have no member in common! It means that such teams are distinct: there is at least a member in one team that is not in the other one.
Then the required probability is $$P(A_3\cup A_4 \cup A_5)=P(A_3)+P(A_4)+P(A_5).$$ What is $P(A_k)$? We choose $11$ members out of $15$, where $k$ are selected among the $5$ bowlers and $11-k$ among the remaining $15-5$ members.
• $A_4$ and $A_5$ are mutually exclusive: If a team has EXACTLY $4$ bowlers it has not EXACTLY $5$ bowlers and vice-versa. – Robert Z May 31 '18 at 7:21
• Mutually exclusive does not mean that a team in $A_4$ and another team in $A_5$ have no member in common! It means that such teams are distinct. – Robert Z May 31 '18 at 7:26