Probability
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W01 - HW Solutions

01

(a)

(1) Count outcomes:

Since there are 6 possible results of rolling a die, and possible results of a coin flip, our sample space has 12 elements.

(2) Use set builder notation to describe the sample space :

(b)

Note that the number of possible events amount to counting how many subsets there are of . In other words, we are asked to compute .

Compute .

02

(a)

(1) Divide the sample space into two disjoint sets:

Denote as the sample space where the result of the die is even.

Denote as the sample space where the result of the die is odd.

.

(2) Describe .

There are even numbers on a die, and possible results of each coin flip. Since coin flips are independent has elements.

(3) Write using set builder notation.

(4) Describe .

There are odd numbers on a die, and 2 possible results of a coin flip. has elements.

(5) Write using set builder notation.

(6) Describe .

As above, .

Since and are disjoint, .

(b)

(1) Note that the number of possible events amount to counting how many subsets there are of . In other words, we are asked to compute .

(2) Compute .

03

(a) Since only the third car is broken, and the other three cars can have any status, the relevant set is

(b) In this case, either all cars are good or all cars are broken. Therefore, the relevant set is

(c) In this scenario, the only combination not in the relevant set is ‘GGGG’. Therefore,

(d) In this scenario, given two cars and , . Therefore,

04

(1) is computed by directly applying the inclusion-exclusion principle.

(2)

(3)

(4) We can express as . Therefore,

(5)

05

(1) State the inclusion-exclusion principle.

(2) Examine possibilities based on given values.

Given that is and is , we have that .

Since , we know that .

Therefore, .

06

(a)

(b)

07

(1) Describe the sample size of this experiment.

(2) Find the probability that at least two heads appear.

The sequences of flips that contain at least two heads are , , , .

We know that , thus

08

(1) We are asked to compute . Set up the conditional probability formula.

(2) We have from the table that and . Therefore,

(3) For 1st-year students, we have

09

(1) Let be the outcome of the first die and be the outcome of the second die. We are asked to compute

(2) Compute individual probabilities.

There is only one combination out of the 36 possible combinations of two dice rolls in which at least 1 die rolls a 5 and both sum up to 10 (5, 5).

There are combinations of dice rolls in which at least one is a .

(3) Plug into formula.

10

(1) Let be the outcome of the first die and be the outcome of the second die. We are asked to compute

(2) Compute individual probabilities.

There are combinations in which at least one die rolled a . Since one of these combinations is , we have 10 combinations in which the outcomes are unequal.

There are combinations in which the outcome of the two dice differ.

(3) Plug into formula.

15

(1) Set up conditional probability formula.

Solve for .

(2) Plug in given values.

(3) Set up conditional probability formula.

Solve for .

(4) Plug in given values.

(5) Use inclusion-exclusion principle to find .