A survey of 4826 randomly selected young adults (aged 19 to 25) asked, "What do you think

A survey of 4826 randomly selected young adults (aged 19 to 25) asked, "What do you think

Question
Two-way tables
asked 2020-12-21

A survey of 4826 randomly selected young adults (aged 19 to 25) asked, "What do you think are the chances you will have much more than a middle-class income at age 30?" The two-way table summarizes the responses.
\(\begin{array}{c|cc|c} &\text { Female } & \text { Male } & \text { Total } \\ \hline \text { Almost no chance } & 96 & 98 & 194 \\ \hline \text { Some chance but } \ \text { probably not } & 426 & 286 & 712 \\\hline \text { A 50-50 chance } & 696 & 720 & 1416 \\ \hline \text { A good chance } & 663 & 758 & 1421 \\ \hline \text { Almost certain } & 486 & 597 & 1083 \\ \hline \text { Total } & 2367 & 2459 & 4826 \end{array}\)
Choose a survey respondent at random. Define events G: a good chance, M: male, and N: almost no chance. Find \(P(C∣M)\). Interpret this value in context.

Answers (1)

2020-12-22
Definitions
Definition conditional probability:
\(P(B|A)=\frac{P(A\cap B)}{P(A)}=\frac{\text{P(A and B)}}{P(A)}\)
SOLUTION
\(\begin{array}{c|cc|c} &\text { Female } & \text { Male } & \text { Total } \\ \hline \text { Almost no chance } & 96 & 98 & 194 \\ \hline \text { Some chance but } \ \text { probably not } & 426 & 286 & 712 \\\hline \text { A 50-50 chance } & 696 & 720 & 1416 \\ \hline \text { A good chance } & 663 & 758 & 1421 \\ \hline \text { Almost certain } & 486 & 597 & 1083 \\ \hline \text { Total } & 2367 & 2459 & 4826 \end{array}\)
G=Good chance
M=Male
We note that the table contains information about 4826 young adults (given in the bottom right corner of the table).
Moreover, 2459 of the 4826 young adults are male, becatise 2450 is mentioned in the row "Total” and in the column ”Male” of the table.
The probability is the number of favorable outcomes divided by the number of possible outcomes:
\(P(M)=\frac{\text{# of favorable outcomes}}{\text{of possible outcomes}}=\frac{2459}{4826}\)
Next, we note that. 758 of the 4826 children are from England and prefer Telepathy, because 758 is mentioned in the row ”A good chance” and in the column "Male” of the given table.
\(P(G \text{and} M)=\frac{\text{# of favorable outcomes}}{\text{of possible outcomes}}=\frac{758}{4826}\)
Use the definition of conditional probability:
\(P(G|M)=\frac{P(G \text{and} M)}{P(M)} = \frac{\frac{758}{4826}}{\frac{2459}{4826}}=\frac{758}{2459}\approx0.3083=30.83\%\)
30.83% of the male young adults think there is a good chance that they will have much more than a middle-class income at age 30.
\(\frac{758}{2459}\approx0.3083=30.83\%\)
0

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