Question

In: Statistics and Probability

Suppose that 8 Secret Agents are gathered together in a secret room and that 2 of...

Suppose that 8 Secret Agents are gathered together in a secret room and that 2 of the agents are "double-agents" (they also work for another spy agency). If 4 agents are selected for a special, top-secret mission, what is the probability that a double-agent (at least one) is selected for the mission?

Solutions

Expert Solution

Number of secret agents in a secret room = 8

Number of double agents in the room = 2

Number of non-double agents in the room = 8-2 =6

Number of agents selected for a special, top secret mission = 4

Event of selecting at least one double-agent is selected

= Event of exactly one double-agent is selected and three non-double agents are selected OR

Event of both the double-agents are selected and two non-double agents are selected

Probability that a double-agent (at least one) is selected for the mission

= Probability of exactly one double-agent is selected and three non-double agents are selected +

Probability of both the double-agents are selected and two non-double agents are selected

Probability of exactly one double-agent is selected and three non-double agents are selected  

= Number of ways of selecting 1 double agent from 2 double agents x Number of ways of selecting 3 non-double agents from 6 non-double agents / Number of ways of selecting 4 agents from 8 agents

=

Probability of exactly one double-agent is selected and three non-double agents are selected = 4/7

Probability of both the double-agents are selected and two non-double agents are selected

= Number of ways of selecting 2 double agent from 2 double agents x Number of ways of selecting 2 non-double agents from 6 non-double agents / Number of ways of selecting 4 agents from 10 agents

=

Probability of both the double-agents are selected and two non-double agents are selected = 3/14

Probability that a double-agent (at least one) is selected for the mission

= Probability of exactly one double-agent is selected and three non-double agents are selected +

Probability of both the double-agents are selected and two non-double agents are selected

= (4/7) + (3/14) = 11/14

Probability that a double-agent (at least one) is selected for the mission = 11/14 = 0.785714286


orchestra answered 1 year ago
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