Respuesta :
From the information available, the mixture will test negative if all 6 samples are negative.
The probability of each is independent of the other for all 6 samples.
The probability of a sample testing positive is 0.11. That means the probability of a sample testing negative would be
[tex]\begin{gathered} P\lbrack neg\rbrack=1-P\lbrack pos\rbrack \\ P\lbrack\text{neg\rbrack}=1-0.11 \\ P\lbrack\text{neg\rbrack}=0.89 \end{gathered}[/tex]However, for all 6 samples, the probability of having a negative result would be a product of probabilities, that is;
[tex]\begin{gathered} P\lbrack tests\text{ negative}\rbrack=0.89\times0.89\times0.89\times0.89\times0.89\times0.89 \\ P\lbrack\text{tests negative}\rbrack=0.89^6 \\ P\lbrack\text{tests negative\rbrack}=0.4969 \end{gathered}[/tex]Therefore if we have the probability of the mixture testing negative as
[tex]P_{\text{neg}}=0.4969[/tex]The probability of the mixture testing positive would be;
[tex]\begin{gathered} P_{\text{pos}}=1-P_{\text{neg}} \\ P_{\text{pos}}=1-0.4969 \\ P_{\text{pos}}=0.5031 \end{gathered}[/tex]ANSWER:
The probability that the mixture will test positive is 0.5031
Rounded to 2 decimal places,
[tex]P_{\text{pos}}=0.50[/tex]