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 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. Given that the chosen student didn't say "almost no chance," what's the probability tha

Question
Two-way tables
asked 2021-01-19
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.
\(\begin{array} {c|cc|c} & \text { Female } & \text { Male } & \text { Total } \\ \hline \text { Almost no chance } & 96 & 98 & 194 \\ \hline \text { Some chance but probably not } & 426 & 286 & 712 \\ \hline \text { A 50-50 chance } & 696 & 720 & 1416 \\ \hline \text { A good chance } & 663 & 758 & 1421 \\ \hline \text { Almost certain } & 486 & 597 & 1083 \\ \hline \text { Total } & 2367 & 2459 & 4826 \end{array}\)
Choose a survey respondent at random. Define events G: a good chance, M: male, and N: almost no chance. Given that the chosen student didn't say "almost no chance," what's the probability that this person is female? Write your answer as a probability statement using correct symbols for the events.

Answers (1)

2021-01-20
Definitions
Complement rule
\(P(A^{c})=P(\text{not}\ A)=1-P(A)\)
Definition conditional probability:
\(P(B or A)=\frac{P(A \cap B)}{P(A)}=\frac{P(A and B)}{P(A)}\)
Solution
\(\begin{array} {c|cc|c} & \text { Female } & \text { Male } & \text { Total } \\ \hline \text { Almost no chance } & 96 & 98 & 194 \\ \hline \text { Some chance but probably not } & 426 & 286 & 712 \\ \hline \text { A 50-50 chance } & 696 & 720 & 1416 \\ \hline \text { A good chance } & 663 & 758 & 1421 \\ \hline \text { Almost certain } & 486 & 597 & 1083 \\ \hline \text { Total } & 2367 & 2459 & 4826 \end{array}\)
N=Almost no chance
M=Male
We note that the table contains information about 4826 young adults (given in the bottom right corner of the table).
Moreover, 194 of the 4826 young adults have the opinion "Almost no chance” because 194 is mentioned in the row ” Almost no chance” and in the column "Total " of the table. This then implies that 4826-194 = 4632 young adults do not have the opinion of "Almost no chance"
The probability is the number of favorable outcomes divided by the number of possible outcomes:
\(P(N^c)=\frac{\text{# of favorable outcomes}}{\text{of possible outcomes}}=\frac{4632}{4826}\)
Next, we note that 96 of the 4826 young adults are female and have the opinion ” Almost no chance”, because 96 is mentioned in the row ” Almost no chance” and in the column *Female” of the given table. Since there are 2367 females, 2367 - 96 = 2271 of the 4826 young adults did not have the opinion ” Almost no chance.” \(P(M^c \text{and} N^c)=\frac{\text{# of favorable outcomes}}{\text{of possible outcomes}}=\frac{2271}{4826}\)
Use the definition of conditional probability:
\(P(M^c | N^c)=\frac{P(M^c \text{and} N^c)}{P(N^c)}=\frac{\frac{2271}{4826}}{\frac{4632}{4826}}=\frac{2271}{4632}=\frac{757}{1544} \approx 0.4903 = 49.03\%\)
Result \(\frac{757}{1544} \approx 0.4903 = 49.03\%\)
0

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