Question

Combining independent probabilities. You have applied to three schools: University of California at San Francisco (UCSF), Duluth School of Mines (DSM), and Harvard (H). You guess that the probabilities you'll be accepted are \(p(\mathrm{UCSF})=0.10, p(\mathrm{DSM})=0.30\), and \(p(\mathrm{H})=0.50\). Assume that the acceptance events are independent. (a) What is the probability that you get in somewhere (at least one acceptance)? (b) What is the probability that you will be accepted by both Harvard and Duluth?

Short answer

Probability of at least one acceptance: 0.685. Probability of being accepted to both Harvard and Duluth: 0.15.

Step-by-step solution

Identify the given probabilities

Given probabilities are: \( p(\mathrm{UCSF}) = 0.10 \) \( p(\mathrm{DSM}) = 0.30 \) \( p(\mathrm{H}) = 0.50 \)

Calculate the probability of not being accepted at any school

First, calculate the probability of not getting accepted into each school: \( p(\text{not UCSF}) = 1 - p(\mathrm{UCSF}) = 1 - 0.10 = 0.90 \) \( p(\text{not DSM}) = 1 - p(\mathrm{DSM}) = 1 - 0.30 = 0.70 \) \( p(\text{not H}) = 1 - p(\mathrm{H}) = 1 - 0.50 = 0.50 \)

Calculate the probability of not getting accepted at any school

Since the events are independent, the probability of not being accepted to any school is the product of the individual probabilities: \( p(\text{not any school}) = p(\text{not UCSF}) \times p(\text{not DSM}) \times p(\text{not H}) = 0.90 \times 0.70 \times 0.50 = 0.315 \)

Calculate the probability of being accepted to at least one school

The probability of getting in somewhere (at least one acceptance) is the complement of the probability of not getting accepted to any school: \( p(\text{at least one}) = 1 - p(\text{not any school}) = 1 - 0.315 = 0.685 \)

Calculate the probability of being accepted to both Harvard and Duluth

Since the acceptance events are independent, the probability of being accepted to both Harvard and Duluth is the product of the individual probabilities: \( p(\text{Harvard and Duluth}) = p(\mathrm{H}) \times p(\mathrm{DSM}) = 0.50 \times 0.30 = 0.15 \)

Key concepts

Probability

Probability is the measure of the likelihood that a given event will occur. It is often expressed as a number between 0 and 1, where 0 indicates an impossibility and 1 indicates certainty. In any given scenario, probabilities can help predict outcomes. For example, in our scenario, we have the probabilities of being accepted into three different schools: UCSF, DSM, and Harvard. These are given as:

• University of California at San Francisco (UCSF): 0.10
• Duluth School of Mines (DSM): 0.30
• Harvard (H): 0.50

The sum of probabilities for all possible outcomes in a particular event must equal 1. Probabilities provide us with a useful tool for making predictions about uncertain events.

Independent Events

Independent events are those events where the occurrence of one event does not affect the occurrence of another. In our scenario, being accepted into UCSF, DSM, and Harvard are considered independent events. This means that whether you get into UCSF does not change the probability of getting into DSM or Harvard. For example:

• The probability of getting into UCSF is 0.10, and it remains 0.10 regardless of what happens with the other two schools.
• Similarly, the probabilities for DSM and Harvard are 0.30 and 0.50, respectively, and they do not influence each other.

Independence of events allows for the use of the multiplicative rule, making calculations straightforward.

Complement Rule

The complement rule helps us find the probability of an event not happening. The complement of an event A is the event that A does not happen. This rule is handy when it's easier to calculate the probability of the complementary event. The complement rule is expressed as: \[ p(\text{not A}) = 1 - p(\text{A}) \]In our scenario, we used the complement rule to find the probabilities of not getting accepted into each school:

• The probability of not getting into UCSF: \[ p(\text{not UCSF}) = 1 - 0.10 = 0.90 \]
• The probability of not getting into DSM: \[ p(\text{not DSM}) = 1 - 0.30 = 0.70 \]
• The probability of not getting into Harvard: \[ p(\text{not H}) = 1 - 0.50 = 0.50 \]

The complement rule simplifies the process of finding the probability of the opposite event, which can then be used in further calculations.

Multiplicative Rule

The multiplicative rule is used to determine the probability of two or more independent events occurring together. For independent events A and B, the rule is given by:\[ p(A \text{ and } B) = p(A) \times p(B) \]In our scenario, to find the probability of not getting accepted into any school, we used the multiplicative rule for the probabilities of not being accepted into each of the three schools:

• \[ p(\text{not any school}) = p(\text{not UCSF}) \times p(\text{not DSM}) \times p(\text{not H}) = 0.90 \times 0.70 \times 0.50 = 0.315 \]

Similarly, to find the probability of being accepted to both Harvard and Duluth simultaneously, we multiplied their individual acceptance probabilities:

• \[ p(\text{Harvard and Duluth}) = p(\text{H}) \times p(\text{DSM}) = 0.50 \times 0.30 = 0.15 \]

The multiplicative rule is crucial in scenarios involving independent events, such as determining joint probabilities.