In a study of pleas and prison sentences, it is found that 40% of the subjects studied
were sent to prison. Among those sent to prison, 20% chose to plead guilty. Among
those not sent to prison, 10% chose to plead guilty.
3.3.a (0.5 pt) If a study subject is randomly selected, what is the probability that this
person was not sent to prison?
3.3.b (0.5 pt) If a study subject is randomly selected, what is the probability that this
study subject entered a guilty plea?
3.3.c (0.5 pt) If a study subject is randomly selected and it is then found that this
subject entered a guilty plea, �nd the probability that this person was not sent
to prison.
25
.0755
To solve these questions, we can use the concept of conditional probability. Conditional probability calculates the probability of an event happening given that another event has already occurred.
Let's solve each question step by step:
3.3.a To find the probability that a randomly selected study subject was not sent to prison, we need to subtract the probability of being sent to prison from 1. So the equation becomes:
P(not sent to prison) = 1 - P(sent to prison)
Given that 40% of the subjects were sent to prison, the probability of being sent to prison is 0.40.
P(not sent to prison) = 1 - 0.40 = 0.60
Therefore, the probability that a randomly selected study subject was not sent to prison is 0.60 or 60%.
3.3.b To find the probability that a randomly selected study subject entered a guilty plea, we need to consider both the subjects sent to prison who pleaded guilty and the subjects not sent to prison who pleaded guilty.
P(guilty plea) = P(guilty plea ∩ sent to prison) + P(guilty plea ∩ not sent to prison)
Given that 20% of the subjects sent to prison pleaded guilty and 10% of the subjects not sent to prison pleaded guilty, the equation becomes:
P(guilty plea) = 0.20 + 0.10 = 0.30
Therefore, the probability that a randomly selected study subject entered a guilty plea is 0.30 or 30%.
3.3.c To find the probability that a randomly selected study subject was not sent to prison given that they entered a guilty plea, we need to use Bayes' theorem. Bayes' theorem allows us to calculate the probability of an event happening given that another event has already occurred.
P(not sent to prison | guilty plea) = (P(guilty plea | not sent to prison) * P(not sent to prison)) / P(guilty plea)
Given that 10% of the subjects not sent to prison pleaded guilty, and we already calculated that P(guilty plea) is 0.30, the equation becomes:
P(not sent to prison | guilty plea) = (0.10 * 0.60) / 0.30 = 0.20
Therefore, the probability that a randomly selected study subject was not sent to prison given that they entered a guilty plea is 0.20 or 20%.