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Transcript
Probability Rules:
Independent Events - events where the occurrence or nonoccurrence of one does not change the
probability that the other will occur.
For example: You select a card at random, record it, and then place it back in the deck.
Since you replaced it, the probabilities when you select the 2nd card do not change. This is
called independence. However, if you did not place the card back in the deck, the probabilities
of your second selection would change.
Dependent Events - when the outcome of the first event changes the probability of the next
event.
Can you think of any examples that are independent or dependent?
Does the independence or dependence of an event matter?
Independence or dependence determines the way we compute probability of two events
happening together.
For Independent Events, P(A and B) = P(A) · P(B).
For Dependent Events, P(A and B) = P(A) · P(B, given that A has occurred).
"
"
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, P(A and B) = P(B) · P(A, given that B has occurred).
If the events are dependent, we have to take into account the changes in the probability of one
event caused by the occurrence of the other event. The notation P(A, given B) denotes the
probability that event A will occur, given that event B has occurred. We call this conditional
probability.
*** View Example #'s 3 & 4 (text p.199 - 201).
Probability of A or B
The condition A or B is satisfied by any of the following:
1) Any outcome of A occurs.
2) Any outcome of B occurs.
3) Any outcome in A and B occurs.
Example: If you want to compute the probability of drawing an ace or a king on 2 consecutive
cards, you would just add the two probabilities together.
It is important to distinguish between the "or" combinations and the "and" combinations
because we apply different rules to compute the probabilities.
*** View Guided Exercise #6 (text p. 203).
We will use the above formulas, along with the multiplication rules of probability to solve
problems.
It is a good idea to be familiar with:
1) Dice
2) Deck of 52 Cards
Examples:
1) The probability of throwing two fours on a single toss of a pair of dice is
a) 1/6
b) 1/3
c) 1/12
d) 1/36
2) If two coins are tossed the probability of getting two tails is
a) 1/8
b) 1/3
c) 1/4
d) 1/2
3) If two cards are drawn from a standard deck of 52 cards without replacement, what is the
probability that both cards are fives?
a) 4/52 · 3/52 b) 5/52 · 4/51 c) 1/4 · 1/3
d) 2/52
4) From a deck of 52 cards, two cards are randomly drawn without replacement. What is the
probability of drawing two hearts?
a) 13/52 · 12/51
b) 13/52 · 13/51
c) 2/52
d) 13/52 · 13/51
5) If two cards are drawn from a standard deck of 52 cards without replacement, what is the
probability that both cards will be black aces?
a) 2/52 · 2/51
b) 4/52 · 3/51
c) 4/52 · 4/51
d) 2/52 · 1/51
6) If 2 cards are dealt randomly from a standard deck of 52 cards, what is the probability that
they are both red queens?
a) 2/52 · 1/51
b) 2/26
c) 4/52 · 31/51
d) 2/52
7) From a standard deck of 52 cards, two cards are drawn at random without replacement. What
is the probability that both cards drawn are aces?
a) 12/2,652
b) 4/2,652
c) 4/52
d) 6/2,652
8) A gumball machine contains six yellow gumballs and five orange gumballs. What is the
probability of obtaining, at random and without replacement, two yellow gumballs?
a) 30/110
b) 36/121
c) 30/121
d) 36/110
9) A bag of marbles contains three blue, one black, and four yellow marbles. If two marbles are
chosen at random without replacement, what is the probability that both marbles will be yellow?
a) 3/14
b) 7/56
c) 1/3
d) 1/4
10) A pencil holder contains only six blue pencils and three red pencils. If two pencils are drawn
at random, what is the probability both are blue?
a) 6/9
b) 30/72
c) 2/9
d) 30/81
11) An urn contains three red marbles and four blue marbles. One marble is selected at random,
its color is noted, and it is returned to the urn. Another marble is then selected. What is the
probability the second marble drawn is red?
a) 3/7
b) 1/7
c) 3/6
d) 2/7
12) When a fair die is tossed, what is the probability of getting a number divisible by both 2 and
3?
13) What is the probability that a student chosen at random from this class has brown hair?
14) If two coins are tossed, what is the probability of getting one head and one tail?
15) A fair die and a fair coin are tossed. What is the probability of obtaining an 8 on the die and
a head on the coin?
16) If a fair coin is tossed, what is the probability of getting three tails?
17) At a certain point in a game, Nancy needs to roll either a 2 or a 3 on a die to win. What is the
probability of her rolling either a 2 or a 3 on one roll of a fair six-sided die?
18) What is the probability that a student chosen at random from this class has red or brown hair?
19) A fair die and a fair coin are tossed. What is the probability of obtaining a 5 on the die and a
tail on the coin?
20) A number is selected at random from the set {1,3,5,8,11,15}. What is the probability the
number is greater than 8 or less than 3?
21) Eight of Mrs. Smith's students weigh 79, 60, 80, 50, 55, 100, 80, and 72. If one of the
students is picked at random, find the probability that the student's weight will be greater than the
mean.
22) Eight of Mrs. Smith's students weigh 79, 60, 80, 55, 50, 100, 80, and 72 pounds. If one of
the students is picked at random, find the probability that the student's weight will be less than
the mode.
23) A rectangular floor uses 50 tiles for its length and 20 tiles for its width. Ten of the tiles in
this floor are cracked. In an inspection, one tile is selected at random. What is the probability
that the tile is cracked?
24) Adam has a bag containing four yellow gumdrops and one red gumdrop. He will eat one of
the gumdrops, and a few minutes later, he will eat a second gumdrop.
a) What is the probability Adam will eat a yellow gumdrop first and a red gumdrop
second?
b) What is the probability that Adam will eat two yellow gumdrops?
c) What is the probability Adam will eat two gumdrop having different colors?
d) What is the probability Adam will eat two red gumdrops?
Mutually Exclusive Events - events that cannot occur together.
- events that do not have any outcomes in common.
- P(A and b) = 0
For mutually exclusive events A and B, P (A or B) = P(A) + P(B).
If the events are not mutually exclusive, we use a more general formula, which is the addition
rule for any events A and B.
P(A or B) = P(A) + P(B) - P(A and B)
***If you are unsure as to which formula to use, always use the 2nd formula. This will
give you the correct answer regardless.
*** View Guided Exercise #7 & 8 (text p.205 - 206)
Combination of Several Events
The addition rule for mutually exclusive events can be extended so that it applies to the
situation in which we have more than two events that are mutually exclusive to all other events.
*** View Example #5 p. 206.
*** View Section 4.2 Problems (text p. 210 – 221).