Question

In a blood testing procedure, blood samples from 6 people are combined into one mixture. The mixture will only test negative if all the individual samples are negative. If the probability that an individual sample tests positive is 0.11. What is the probability that the mixture will test positive?

Answer 1

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Answer 2

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

`\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}`

However, for all 6 samples, the probability of having a negative result would be a product of probabilities, that is;

`\begin{gathered} P\lbrack tests\text{ negative}\rbrack=0.89*0.89*0.89*0.89*0.89*0.89 \\ P\lbrack\text{tests negative}\rbrack=0.89^6 \\ P\lbrack\text{tests negative\rbrack}=0.4969 \end{gathered}`

Therefore if we have the probability of the mixture testing negative as

`P_{\text{neg}}=0.4969`

The probability of the mixture testing positive would be;

`\begin{gathered} P_{\text{pos}}=1-P_{\text{neg}} \\ P_{\text{pos}}=1-0.4969 \\ P_{\text{pos}}=0.5031 \end{gathered}`

ANSWER:

The probability that the mixture will test positive is 0.5031

Rounded to 2 decimal places,

`P_{\text{pos}}=0.50`