 # Discrete Math Questions

## Question

*Two faces of a six-sided die are painted red, two are painted blue, two are yellow. The die is rolled 3 times and the colors that appear face up on the 1st, second and third rolls are recorded.
(a) find the probability of the event that exactly one of the colors that appears face up is red.
(b) find the probability of the even that at least one of the colors that appears face up is red.

*An urn contains two blue balls( B_1, B_2), and 3 white balls(W_1, W_2, W_3). One ball is drawn, its color is recorded, and it i s replaced in the urn. Then another ball is drawn and its color is recorded.
(a) Let B_1 W_2 denote the outcome that the first ball drawn is B_1 and the 2nd is W_2. Because the first ball is replaced before the 2nd ball ids drawn, the outcomes of the experiment are equally likely. List all 25 possible outcomes.
(b) consider the event that the first ball that is drawn is blue. List all outcomes in the event. What is the probability of the event.
(c) Consider the event that only white balls are drawn. list all outcomes in the event. what is the probability of the event?

*Answer the following:
(a) how many positive 3-digit integers are multiples of 6?
(b) what is the probability that a randomly chosen positive 3 digit integer is a multiple of 6.
(c) what is the probability that a randomly chosen positive three-digit integer is a multiple of 7?

*A combination lock requires three selections of numbers each from 1 through 30.
(a) how many different combinations are possible?
(b) suppose the locks are constructed in such a way that no number may be used twice. how many different combinations are there?

*Answer the following:
(a) how many integers are there from 1000 through 9999?
(b) how many odd ints are there from 1000 through 9999
(d) how many odd ints are from 1000 through 9999 that have distinct digits?
(e) what is the probability that a randomly chosen 4 digit integer has distinct digits? has distinct digits and is odd?

*Six people attend the theater together and sit in a row with exactly 6 seats.
(a) how many ways can they be seated together in the row?
(b) suppose one of the six is a doctor who must sit on the aisle in case she is paged. How many ways can the people be seated together in the row with the doctor in an aisle seat.
(c) suppose the six people consist of three married couples and each couple wants to sit together with the husband on the left. How many ways can the six be seated together in the row?

*In another state, all license plates consist of from four to six symbols chosen from 26 letters of the alphabet together with the ten digits 0-9.
(a) how many license plates are possible if repetition of symbols is allowed?
(b) how many license plates do not contain any repeated symbol?
(c) how many license plates have at least one repeated symbol?
(d) what is the probability tghata license plate chosen at random has a repeated symbol?

*Answer the following:
(a) how many integers from 1 through 1000 are multiples of 2 or multiple of 9?
(b) suppose an integer from 1 through 1000 is chosen at random. use the result of part(a) to find the probability that the integer is multiple of 2 or a multiple of 9.
(c) how many integers from 1 through 1000 are neither multiples of 2 nor multiples of 9?

*Assuming that all years have 365 days and all birthdays occur with equal probability, how large must n be so that in any randomly chosen group of n people, the probability that two or more have the same birthday is at least 1/2? (this is called the birthday problem).

## Solution Preview

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Two faces of a six-sided die ar epaiinted red, two are painted blue, two are yello. The die is rolled 3 times and the colors that appear face up on the 1st, second and third rolls are recorded.

We have   2R 2B 2Y
So, in 1 throw,
P (R)= 2/6 =0.33

a>
So, in 3 throws,
P(only 1 R)= 3C1 (0.33)1(1-0.33)3-1 = 0.444

b>
We need P(R=1,2,3)
P(R=2) =3C2 (0.33)2(1-0.33)3-2 = 0.222
P(R=3) =3C3 (0.33)3(1-0.33)3-3 = 0.037
So, required probability of at least 1 red in 3 throws = 0.444+0.222+0.037 = 0.703...
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