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Question: From a group of two men and three women, two persons are selected. Describe the sample space of the ...

From a group of two men and three women, two persons are selected. Describe the sample space of the experiment. If EE is the event in which one man and one woman are selected, then which are the cases favorable to EE ?

Explanation

Solution

Hint : Sample space is the collection of all possible outcomes. Write two sets. One for men and one for women. Then combine the elements in those two sets in the collection of two people at a time to form all possible outcomes for sample space. Then find the elements in the sample space which have one man and one woman, to form the favorable outcomes for E.

Complete step-by-step answer :
Let M be the set containing two men. Then, M is written as
M=M1,M2M = \\{ {M_1},{M_2}\\}
Let W be the set containing two women. Then, W is written as
W=W1,W2,W3W = \\{ {W_1},{W_2},{W_3}\\}
Let S be the sample space. Then, by the property of set theory, sample space for selecting two people can be written as
S=⇒(M1,M2),(M1,W1),(M1,W2),(M1,W3),(M2,W1),(M2,W2),(M2,W3),(W1,W2),(W1,W3),(W2,W3)S = \\{ \Rightarrow ({M_1},{M_2}),({M_1},{W_1}),({M_1},{W_2}),({M_1},{W_3}),({M_2},{W_1}),({M_2},{W_2}),({M_2},{W_3}),({W_1},{W_2}),({W_1},{W_3}),({W_2},{W_3})\\}
⇒n(S)=10\Rightarrow n(S) = 10
Thus the sample space will have 10 elements.
Let E be the event of selecting one man and one woman. Then E will be the subset of S in which one man and one woman is given as an element.
Thus, E can be written as
E=⇒(M1,W1),(M1,W2),(M1,W3),(M2,W1),(M2,W2),(M2,W3)E = \\{ \Rightarrow ({M_1},{W_1}),({M_1},{W_2}),({M_1},{W_3}),({M_2},{W_1}),({M_2},{W_2}),({M_2},{W_3})\\}
⇒n(E)=6\Rightarrow n(E) = 6
Thus the class favorable to E will be the one explained above. And the number of elements in the class will be 6.

Note : We wrote (M1,W1)({M_1},{W_1}) as an element but we did not write (W1,M1)({W_1},{M_1}) as an element. Writing (W1,M1)({W_1},{M_1}) instead of (M1,W1)({M_1},{W_1}) would not have been wrong. We can write any one of them. But we cannot write both of them. Because, when it comes to selection, it does not matter who we select first. Man or woman. Therefore, (M1,W1)({M_1},{W_1}) and (W1,M1)({W_1},{M_1}) are the same in terms of selection. And in set theory, we do not write the same element again.