Probability Flashcards
probability
ratio P = nb of successes / total nb of outcomes impossibility P = 0 certainty P = 1 0 < P < 1 P(not A) = 1 - P(A)
mutually exclusive
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)
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?
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
independent events
no effect on each other
the outcome of one has no influence on the outcome of the other
with replacement
choices are made from the same pool and are independent of the previous ones
P(A and B) = P(A)*P(B)
without replacement
choices are made under different condition, each choice changes the probability for all successive choices, which are not independent
What is the probability of tossing three coins and getting HHH?
1/2 * 1/2 * 1/2 = 1/8
What is the probability of rolling two six-sided dice and getting “snake eyes” (each die = 1)?
1/6 * 1/6 = 1/36
Three cards are selected from a full deck, each time with replacement. What is the probability of selecting three Spades in a row?
1/4 * 1/4 * 1/4 = 1/64
Suppose A and B are independent. If P(A) = 0.6 and P(B) = 0.8, what does P(A or B) equal?
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
conditional probability
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
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?
P(1=G) = 5/12 P(2=G) = 4/11 P = 5/12 * 4/11 = 5/33
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?
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
Three fair coins are flipped. What is the probability of getting exactly two heads?
HHT
HTH
THH
3(1.2 * 1.2 * 1.2) = 3(1/8) = 3/8
Ten dice, each fair with six sides, are rolled simultaneously. What is the probability of getting exactly two fives among them?
10C2 * ( (1/6)² * (5/6)^8 )
General binomial formula
p = probability of success on one trial n = nb of trials r = nb of successes
P = (nCr) * (p^r) * [(1 - p)^(n - r)]
complement rule shortcut
the complement of “at least one” is “none” and we can say
P(at least one success) = 1 - P(zero successes)
We roll one fair six-sided die eight times. What is the probability that we will roll at least one six?
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
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?
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
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?
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
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”
use the algebraic rules
if the full list is very short (fewer than 10)
use the listing technique
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.)
use the counting techniques
if “at least”
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)
