The probability of surviving a certain transplant operation is 0.55. If a patient survives the operation, the probability that his or her body will reject the transplant within a month is 0.20. What is the probability of surviving both of these critical stages?

#### Solution

Let $S$ be the event that patient survives transplant. Let $R$ be the event that patients body reject the transplant.

The probability of surviving a certain transplant operation is 0.55, i.e., `$P(S) = 0.55$`

.

If a patient survives the operation, the probability that his or her body will reject the transplant within a month is 0.20, i.e., `$P(R|S) =0.20$`

.

Thus `$P(R^\prime |S) = 0.80$`

.

The probability of surviving both of these critical stages is

` $$ \begin{aligned} P(S\cap R^\prime) &= P(R^\prime |S) P(S)\\ &= 0.80\times 0.55\\ &= 0.44 \end{aligned} $$ `

#### Further Reading

- Statistics
- Descriptive Statistics
- Probability Theory
- Probability Distribution
- Hypothesis Testing
- Confidence interval
- Sample size determination
- Non-parametric Tests
- Correlation Regression
- Statistics Calculators