01
Counting outcomes - permutations and combinations
In a lottery, five distinct numbers are picked at random from
. How many possible outcomes are there: (a) If we care about the order of numbers.
(b) If the order does not matter.
Solution
10
(a)
If the order matters, then we are dealing with a permutation. We want
distinct numbers, so
(b)
If the order matters, then we are dealing with a combination. We want
distinct numbers, so Link to original
02
Wisconsin flag 2 of 3 days
A kindergarten class hangs a random state flag (50 flags) on the wall every day. What is the probability that two days out of three given days have Wisconsin’s flag?
Solution
11
(1) Consider how the days are chosen.
Since we just want two days out three given days, it is unordered, so we account for this with the
term.
(2) Consider the probability that Wisconsin’s flag is hung on the first two days, and not the third.
The probability that Wisconsin’s flag is hung up is
, and the probability it’s any other flag is . Therefore, the desired probability is
.
(3) Combine terms.
Link to original
03
Drawing balls of distinct color
A bin contains 3 green and 4 yellow balls. Two balls are drawn out.
What is the probability that they are different colors?
Solution
12
(1) Define relevant events.
If two balls of different colors are drawn out, then we choose one of each color.
Since order doesn’t matter, our sample space consists of all the ways we choose 2 balls out of 7.
(2) Compute probability.
Link to original
04
Rolling two dice
Two dice are rolled. Find the probabilities of the following events:
, the event that the sum is 10 , the event that the sum is 12 , the event that the two numbers are equal
Solution
01
(1) Consider the total number of outcomes.
Since there are two dice being rolled, there are 36 total outcomes.
(2) Consider the total number of desired outcomes for
. There are 3 total desired outcomes,
, , (6, 4).
(3) Use formula to find
.
(4) Consider the total number of desired outcomes for
. There is only 1 desired outcome,
.
(5) Use formula to find
.
(6) Consider the total number of desired outcomes for
. There are 6 total desired outcomes:
, , , , , Use formula to find
. Link to original
05
Binomial - Repeated coin flips
A coin is flipped 7 times and the sequence of results recorded as an outcome.
(a) How many possible outcomes have exactly 3 heads?
(b) How many possible outcomes have at least 3 heads?
Solution
12
(a)
Out of
trials, we choose of them to be heads. Thus,
(b)
Out of 7 trials, we choose at least 3 of them to be heads.
Using summation notation, we get
Link to original
06
Multinomial - Colored marbles in a line
How many ways are there to line up 10 colored marbles (2 red, 3 white, 5 blue), assuming you cannot distinguish marbles of the same color?
Solution
03
Use the multinomial coefficient.
Use that
. We have three bins,
, , and . Thus,
Link to original
07
Multinomial - Many rolls of a die
Roll a die 100 times.
(a) What is the probability that you rolled exactly 16 ones and 17 twos? (No need to simplify your answer.) Hint: use three bins. What are the bins?
(b) Using summation notation, write down a formula for the probability of rolling exactly 25 ones and at least 50 twos.
For this problem, use “desired outcomes over total outcomes” (simple counting), not repeated trials theory (next section).
Solution
04
(a)
(1) Consider total number of outcomes.
You roll a die
times, so there are total outcomes.
(2) Consider total number of desired outcomes.
We choose 16 of the 100 to be ones.
We choose 17 of the remaining 100 - 16 = 84 to be twos.
The other
can be any of . So, there are remaining outcomes.
(3) Set up formula.
(b)
(1) Consider total number of desired outcomes.
We choose 25 of the 100 to be ones.
Out of the remaining 100 - 25 = 75 rolls, we choose at least 50 to be two.
The remaining
rolls, where , can be any of , , , . So, there are remaining outcomes.
(2) Set up formula.
Link to original
08
Counting license plates
A license plate must consist of 3 letters followed by 4 digits. Assuming we choose from 26 letters, A-Z, and 10 digits, 0-9, how many different license plates could be created if:
(a) Letters and numbers cannot be repeated.
(b) Letters and numbers can be repeated except that there must be exactly two 9’s.
Solution
06
(a) Letters in
ways Digits in ways For each 3 letter arrangement, there are 5040 digit arrangements. Total number of ways (b) Letters in
ways Decide the spots for the two 9’s first: Remaining 2 digits in ways Total number of ways: Link to original
09
Counting passwords
Suppose a password must be created using 5 letters and 6 digits. (There are 26 letters, a-z, and 10 digits, 0-9.) No letter or digit may be repeated.
(a) How many unique passwords can be created if the letters must come first and the digits last?
(b) How many unique passwords can be created if the 5 letters and 6 digits can appear in any order?
Solution
13
(a)
(b)
Link to original