The espresso project you are heading up will do well with probability 0.80. Given that it does well, you believe that the herbal tea project is likely to do well with probability 0.70.However, if the espresso project does not do well, then the herbal tea project has only a
25% chance of doing well.
Find the probability that both projects do well?
I know that we must find
P(espresso and tea) = P(espresso) x P(espresso given tea)
P(espresso and tea) = 0.80 x P(espresso given tea)
I don't know how to find P(espresso given tea) ? please help
thank you,
To find P(espresso given tea), we can use conditional probability. Given that the espresso project does well, you believe that the herbal tea project is likely to do well with probability 0.70. However, if the espresso project does not do well, then the herbal tea project has only a 25% chance of doing well.
Let's break it down into two cases:
Case 1: The espresso project does well.
In this case, we know that the probability of the herbal tea project doing well is 0.70.
Case 2: The espresso project does not do well.
In this case, we know that the probability of the herbal tea project doing well is 0.25.
Now, we need to determine the overall probability that both projects do well. We can use the law of total probability, which states that the probability of an event can be calculated by summing up the probabilities of that event based on different conditions.
So, to find P(espresso given tea), we need to calculate the weighted average of the probabilities in these two cases, using the probability of each case as the weight:
P(espresso given tea) = P(espresso does well and tea does well) / P(tea does well)
P(espresso given tea) = (P(espresso does well and tea does well in case 1) + P(espresso does well and tea does well in case 2)) / (P(tea does well in case 1) + P(tea does well in case 2))
We have already been given the probabilities for each case. So, we can substitute the values:
P(espresso given tea) = (0.80 * 0.70 + 0 * 0.25) / (0.80 + 0)
Simplifying further:
P(espresso given tea) = (0.56 + 0) / 0.80
P(espresso given tea) = 0.56 / 0.80
P(espresso given tea) = 0.70
Now, substitute this value back into the equation:
P(espresso and tea) = P(espresso) * P(espresso given tea)
P(espresso and tea) = 0.80 * 0.70
P(espresso and tea) = 0.56
Therefore, the probability that both projects do well (espresso and tea) is 0.56 or 56%.