Download 5.2 Notes andExercises

Notes on Chapter 5.2 Elementary Probability
A sample space is the set of
all possible outcomes (distinguishable results) from
an experiment (repeatable action).
If S is the sample space, containing outcomes T i, then subsets of S are called events.
A probability on S is a function P that assigns to each event E f S a number P(E) in [0, 1] and
that satisfies:
P( S ) = 1 and P( E c F ) = P(E) + P(F) for disjoint events E and F.
If all outcomes Ti, are equally likely, then P(E) = |E|/ |S|
EXAMPLE 1 Probability of a given types of poker hand.
From the properties of a probability function, we may conclude
that P(i) = 0 and that P(EC ) = 1 ! P(E).
Theorem If P is a probability on a sample space S
then P(i) = 0
and P(EC ) = 1 ! P(E) for event E
and P( E c F ) = P(E) + P(F) ! P( E 1 F )
and for pairwise disjoint events Ei the P(Union of Ei) = the sum of the P(Ei).
EXAMPLE 2 Probability of a union of events
EXAMPLE 3 Probability of 4 edges in a certain graph being a path from s to f.
EXAMPLE 4 Probability of a die problem.
EXAMPLE 5 Probability of a two-dice problem.
EXAMPLE 6 Probability of tossing a coin n times.
EXAMPLE 7 Infinite sample spaces. [Omit].
MAT251 Exercises 5.2, p195. 1 – 11
1. An integer in {1, 2, 3, ..., 25} is selected at random. Find the probability that the number is
a) divisible by 3 l25/3m / 25 = 8/25 = 0.32
b) divisible by 5 l25/5m / 25 = 5/25 = 0.20
c) a prime. |{2, 3, 5, 7, 11, 13, 17, 19, 23}| / 25 = 9/25 = 0.36
2. A letter of the alphabet is selected at random. What is the probability that it is
a vowel {a, e, i, o, u}? 5/26
3. A four-letter word is selected at random from 34 where 3 = {a, b, c, d, e}.
a) What is the probability that the letters in the word are distinct?
b) What is the probability that there are no vowels in the word?
c) What is the probability that the word begins with a vowel?
4. A five-letter word is selected at random from 35 where 3 = {a, b, c}.
a) What is the probability that the letters in the word are distinct? 0
b) What is the probability that there are no vowels in the word?
c) What is the probability that the word begins with a vowel?
5. An urn contains three red and four black balls. A set of three balls is removed at random from
the urn without replacement. Give the probabilities that the three balls are
a) all red
b) all black
d) two red and one black
c) one red and two black
e) What is the sum of parts a) – d)?
6. An urn has three red and two black balls. Two balls are removed at random without
replacement. What is the probability that the two balls are
a) both red?
b) both black?
c) different colors?
MAT251 Exercises 5.2, p196. 1 – 11
7. Suppose that an experiment leads to events A, B, and C with the following probabilities:
P(A) = 0.5, P(B) = 0.8, P(A1B) = 0.4. find
a) P(BC ) = 1 – 0.8 = 0.2
b) P(AcB) = P(A) + P(B) – P(A1B) = 0.5 + 0.8 – 0.4 = 0.9
c) P(AC cBC ) = P( (A1B)C ) 1 – 0.4 = 0.6
8. Suppose that an experiment leads to events A, B, and C with the following probabilities:
P(A) = 0.6 and P(B) = 0.7. Show that P(A1B) $ 0.3.
P(A1B) = P(A) + P(B) – P(AcB) $ 0.6 + 0.7 – 1 = 0.3
9. A five-card poker hand is dealt, as in Example 10, page 187 and Exercise 15, page 189. Find
the probability of getting [where the number of possible hands is 2598960]
a) four of a kind = C(13kinds,1)*C(4suits,4)*C(12kinds,1)*C(4suits,1)/2598960
= 624 / 2598960
b) three of a kind = C(13kinds,1)*C(4suits,3)*C(12kinds,2)*C(4suits,1)*C(4suits,1)/2598960
= 54912 / 2598960
c) an ordinary straight = 10200 / 2598960 by Example 10 part (d)
d) two pairs = 123552 / 2598960 by Example 10 part (c)
e) one pair = C(13ks,1)*C(4ss,2)*C(12ks,3)*C(4ss,1)*C(4ss,1)*C(4ss,1)/ 2598960
= 1098240 / 2598960
10. A poker hand is dealt, as above.
a) What is the probability of getting a hand better than one pair? [any other special hand on p187]
= (4 + 36 + 624 + 3744 + 5108 + 10200 + 54912 + 123552) / 2598960 = 198180 / 2598960
b) What is the probability of getting a pair of jacks or better (pair)? [queens, kings, aces]
4/13 of the 1098240 pairs are jacks, queens, kings, or aces; that is 337920 pairs jacks or
better. Adding to part a) gives 536100 / 2598960
11. A black and a red die are tossed as in Example 5, page 192. What is the probability that
a) the sum of the values is even? P(sum is 2, 4, 6, 8, 10, 12) = (1 + 3 + 5 + 5 + 3 + 1) / 36 = 1/2
b) the number on the red die is bigger than the number on the black die?
[r = red die, b = black die] P({(r, b) : r > b}) = (5+4+3+2+1)/36 = 15/36
c) the number on the red die is twice the number on the black die?
P({(1, 2), (2, 4), (3, 6)}) = 3/36