Probability

Question 1

probability

Answer 1

ratio
P = nb of successes / total nb of outcomes
impossibility P = 0 
certainty P = 1
0 < P < 1
P(not A) = 1 - P(A)

Question 2

mutually exclusive

Answer 2

cannot happen at the same time
P(A and B) = 0

If A and B are not mutually exclusive, then P(A and B) can happen together, and P(A or B) = P(A) + P(B) - P(A and B)

If A and B are mutually exclusive, then P(A or B) = P(A) + P(B)

Question 3

Suppose, in Game M, the probability of outcome A is 0.6, the probability of outcome B is 0.7, and the probability of A or B is 0.9. What is the probability of A and B happening at the same time?

Answer 3

P(A or B) = P(A) + P(B) - P(A and B)
0.9 = 0.6 + 0.7 - P(A and B)
P(A and B) = 0.6 + 0.7 - 0.9 = 0.4

Question 4

independent events

Answer 4

no effect on each other

the outcome of one has no influence on the outcome of the other

Question 5

with replacement

Answer 5

choices are made from the same pool and are independent of the previous ones

P(A and B) = P(A)*P(B)

Question 6

without replacement

Answer 6

choices are made under different condition, each choice changes the probability for all successive choices, which are not independent

Question 7

What is the probability of tossing three coins and getting HHH?

Answer 7

1/2 * 1/2 * 1/2 = 1/8

Question 8

What is the probability of rolling two six-sided dice and getting “snake eyes” (each die = 1)?

Answer 8

1/6 * 1/6 = 1/36

Question 9

Three cards are selected from a full deck, each time with replacement. What is the probability of selecting three Spades in a row?

Answer 9

1/4 * 1/4 * 1/4 = 1/64

Question 10

Suppose A and B are independent. If P(A) = 0.6 and P(B) = 0.8, what does P(A or B) equal?

Answer 10

P(A or B) = P(A) + P(B) - P(A and B)
P(A and B) = P(A)*P(B) = 0.6 * 0.8 = 0.48
P(A or B) = 1.4 - 0.48 = 0.92

Question 11

conditional probability

Answer 11

P(A | B) : what is the probability of A, given B?
happens when A and B are not independent, so whether one happens changes the probability of whether the other happens

P(A and B) = P(B)P(A | B)
P(A and B) = P(A)P(B | A)
mostly used in pb without replacement

Question 12

A box has 5 green balls and 7 red balls. Assume that all balls in the box are equally likely and that the balls are picked without replacement. What is the probability that the first two balls picked are both green?

Answer 12

P(1=G) = 5/12
P(2=G) = 4/11
P = 5/12 * 4/11 = 5/33

Question 13

From a standard shuffled deck of 52 cards, what’s the probability of picking three hearts on the first three cards drawn, if the cards are selected without replacement?

Answer 13

P(1=H) = 1/4 (=13/52)
P(2=H | 1=H) = 12/51 = 4/17
P(3=H | 2=H and 1=H) = 11/50
P = 13/52 * 12/51 * 11/50 = 1/4 * 4/17 * 11/50
P = 11/850

Question 14

Three fair coins are flipped. What is the probability of getting exactly two heads?

Answer 14

HHT
HTH
THH
3(1.2 * 1.2 * 1.2) = 3(1/8) = 3/8

Question 15

Ten dice, each fair with six sides, are rolled simultaneously. What is the probability of getting exactly two fives among them?

Answer 15

10C2 * ( (1/6)² * (5/6)^8 )

Question 16

General binomial formula

Answer 16

p = probability of success on one trial
n = nb of trials
r = nb of successes

P = (nCr) * (p^r) * [(1 - p)^(n - r)]

Question 17

complement rule shortcut

Answer 17

the complement of “at least one” is “none” and we can say

P(at least one success) = 1 - P(zero successes)

Question 18

We roll one fair six-sided die eight times. What is the probability that we will roll at least one six?

Answer 18

the complement of at least one six is zero sixes
the probability of not getting a six in one roll is 5/6
the probability of not getting a single six in eight rolls is (5/6)^8
P(at least one six) = 1 - (5/6)^8

Question 19

In a certain game, in Phase 1 you flip one coin as many as three times. If you flip three tails, you lose. As soon as you get your first head, you advance to Phase 2. In Phase 2, you roll a six-sided die once If you roll a 6, you win. For any other roll, you lose. What is the probability of winning?

Answer 19

P1 up to three coin tosses 
TTT = loss
first H = P2
P2 one die roll
(die = 6) = win
(die = 1-5) = loss

P(win) = P(P1=toP2 and P2,roll6)
P(win) = P(P1=toP2 * P2,roll6)
P(win) = P(P1=toP2 * 1/6)

P(1)
H
TH
TTH
TTT
P(1) = 1 - P(TTT)
P(1) = 1 - (1/2 * 1/2 * 1/2) = 1 - 1/8 = 7/8
P(win) = 7/8 * 1/6 = 7/48

Question 20

A committee of three will be selected from a group of eight employees, including Alice and Bob. What is the probability that the chosen committee of three includes Alice and not Bob?

Answer 20

How many groups of 3 can we make
8C3 = 876 / 321 = 56

How many different pairs can we pick from six (= 8, excluding A and B, so 6)
6C2 = 6*5 / 2 = 15

15/56

Question 21

if the pb gives algebraic expressions
if the items are coins, cards, dice, etc.
if the language of the pb uses the words “mutually exclusive” or “independent”

Answer 21

use the algebraic rules

Question 22

if the full list is very short (fewer than 10)

Answer 22

use the listing technique

Question 23

if the pb involves selection of several elements from a set, with certain restrictions (one is picked, one is not, one is next to another, etc.)

Answer 23

use the counting techniques

Question 24

if “at least”

Answer 24

use the complement rule shortcut
the complement of “at least one” is “none” and we can say
P(at least one success) = 1 - P(zero successes)