Suppose the probability of surviving for 5 years after being diagnosed with a particular cancer is 0.60; and the probability of surviving 10 years is 0.20. If a person survives 5 years, what is the probability that she will survive 10 years.
0.33
To find the probability that a person will survive 10 years given that they have already survived 5 years, we can use conditional probability.
Conditional probability is the probability of an event occurring given that another event has already occurred.
In this case, we want to find the probability of surviving 10 years after surviving 5 years. We can denote this as P(survive 10 | survive 5).
Using the definition of conditional probability, we can write this as:
P(survive 10 | survive 5) = P(survive 10 and survive 5) / P(survive 5)
We know that the probability of surviving 5 years is 0.60, so P(survive 5) = 0.60.
The probability of surviving 10 years is 0.20, so P(survive 10) = 0.20.
To find P(survive 10 and survive 5), we can multiply the probabilities of surviving 10 years and surviving 5 years together:
P(survive 10 and survive 5) = P(survive 10) * P(survive 5) = 0.20 * 0.60 = 0.12
Now we can substitute these values into the conditional probability formula:
P(survive 10 | survive 5) = P(survive 10 and survive 5) / P(survive 5) = 0.12 / 0.60 = 0.2
Therefore, the probability that the person will survive 10 years given that they have already survived 5 years is 0.20 or 20%.