Transcript Probability 3 Lecture

Homework
An experiment consists of rolling a fair
number cube. Find each probability.
1
1. P(rolling an odd number)
21
2. P(rolling a prime number)
2
An experiment consists of rolling two fair
number cubes. Find each probability.
1
3. P(rolling two 3’s) 36
4. P(total shown > 10) 1
12
Warm Up
1. If you roll a number cube, what are the
possible outcomes?
1, 2, 3, 4, 5, or 6
2. Add 1 + 1. 1
12
6
4
3. Add 1 + 2 . 5
2
36
9
Vocabulary
theoretical probability
equally likely
fair
mutually exclusive
disjoint events
Theoretical probability is used to estimate
probabilities by making certain assumptions
about an experiment. Suppose a sample space
has 5 outcomes that are equally likely, that is,
they all have the same probability, x. The
probabilities must add to 1.
x+x+x+x+x=1
5x = 1
x=1
5
A coin, die, or other object is called fair if all
outcomes are equally likely.
Additional Example 1: Calculating Theoretical
Probability
An experiment consists of spinning this spinner
once. Find the probability of each event.
A. P(4)
The spinner is fair, so all 5 outcomes
are equally likely: 1, 2, 3, 4, and 5.
1
P(4) = number of outcomes for 4 =
5
5
Additional Example 1: Calculating Theoretical
Probability
An experiment consists of spinning this spinner
once. Find the probability of each event.
B. P(even number)
There are 2 outcomes in the
event of spinning an even
number: 2 and 4.
P(even number) = number of possible even numbers
5
2
=5
Check It Out! Example 1
An experiment consists of spinning this spinner
once. Find the probability of each event.
A. P(1)
The spinner is fair, so all 5 outcomes
are equally likely: 1, 2, 3, 4, and 5.
1
P(1) = number of outcomes for 1 =
5
5
Check It Out! Example 1
An experiment consists of spinning this spinner
once. Find the probability of each event.
B. P(odd number)
There are 3 outcomes in the
event of spinning an odd
number: 1, 3, and 5.
P(odd number) = number of possible odd numbers
5
3
=5
Additional Example 2: Calculating Probability for a
Fair Number Cube and a Fair Coin
An experiment consists of rolling one fair
number cube and flipping a coin. Find the
probability of the event.
A. Show a sample space that has all outcomes
equally likely.
The outcome of rolling a 5 and flipping heads can
be written as the ordered pair (5, H). There are
12 possible outcomes in the sample space.
1H
1T
2H
2T
3H
3T
4H
4T
5H
5T
6H
6T
Additional Example 2: Calculating Theoretical
Probability for a Fair Coin
An experiment consists of rolling one fair
number cube and flipping a coin. Find the
probability of the event.
B. P(tails)
There are 6 outcomes in the event “flipping tails”:
(1, T), (2, T), (3, T), (4, T), (5, T), and (6, T).
6
1
P(tails) =
=
12
2
Check It Out! Example 2
An experiment consists of flipping two coins.
Find the probability of each event.
A. P(one head & one tail)
There are 2 outcomes in the event “getting one
head and getting one tail”: (H, T) and (T, H).
2
1
P(head and tail) = 4 = 2
Check It Out! Example 2
An experiment consists of flipping two coins.
Find the probability of each event.
B. P(both tails)
There is 1 outcome in the event “both tails”:
(T, T).
1
P(both tails) =
4
Additional Example 3: Altering Probability
Stephany has 2 dimes and 3 nickels. How
many pennies should be added so that the
3
probability of drawing a nickel is 7 ?
Adding pennies to the bag will increase the number
of possible outcomes.
3
= 3
5+x
7
3(5 + x) = 3(7)
Set up a proportion. Let x equal
the number of pennies
Find the cross products.
Additional Example 3 Continued
15 + 3x = 21
–15
– 15
3x = 6
3
3
Multiply.
Subtract 15 from both sides.
Divide both sides by 3.
x= 2
2 pennies should be added to the bag.
Check It Out! Example 3
Carl has 3 green buttons and 4 purple buttons.
How many white buttons should be added so
that the probability of drawing a purple button
2
is 9 ?
Adding buttons to the bag will increase the number
of possible outcomes. Let x equal the number of
white buttons.
4
= 2
7+x
9
2(7 + x) = 9(4)
Set up a proportion. Let x equal
the number of white buttons.
Find the cross products.
Check It Out! Example 3 Continued
14 + 2x = 36
–14
– 14
2x = 22
2
2
Multiply.
Subtract 14 from both sides.
Divide both sides by 2.
x = 11
11 white buttons should be added to the bag.
Two events are mutually exclusive, or
disjoint events, if they cannot both occur in
the same trial of an experiment. For example,
rolling a 5 and an even number on a number
cube are mutually exclusive events because
they cannot both happen at the same time.
Additional Example 4: Finding the Probability of
Mutually Exclusive Events
Suppose you are playing a game in which you
roll two fair number cubes. If you roll a total of
five you will win. If you roll a total of two, you
will lose. If you roll anything else, the game
continues. What is the probability that the
game will end on your next roll?
It is impossible to roll both a total of 5 and a total of
2 at the same time, so the events are mutually
exclusive. Add the probabilities to find the
probability of the game ending on your next flip.
Additional Example 4 Continued
The event “total = 2” consists of 1 outcome, (1, 1), so
P(total = 2) = 1 .
36
The event “total = 5” consists of 4 outcomes, (1, 4),
(2, 3), (4, 1), (3, 2), so P(total = 5) = 4 .
36
P(game ends) = P(total = 2) + P(total = 5)
= 1 +4 = 5
36 36
36
5
The probability the game will end is 36 , or about 14%.
Check It Out! Example 4
Suppose you are playing a game in which you
flip two coins. If you flip both heads you win
and if you flip both tails you lose. If you flip
anything else, the game continues. What is the
probability that the game will end on your next
flip?
It is impossible to flip both heads and tails at the
same time, so the events are mutually exclusive.
Add the probabilities to find the probability of the
game ending on your next flip.
Check It Out! Example 4 Continued
The event “both heads” consists of 1 outcome, (H, H),
so P(both heads) = 1 . The event “both tails” consists of
4
1
1 outcome, (T, T), so P(both tails) =
.
4
P(game ends) = P(both tails) + P(both heads)
=1 + 1
4 4
1
=2
1
The probability that the game will end is 2 , or 50%.
Lesson Quiz
An experiment consists of rolling a fair
number cube. Find each probability.
1
1. P(rolling an odd number)
21
2. P(rolling a prime number)
2
An experiment consists of rolling two fair
number cubes. Find each probability.
1
3. P(rolling two 3’s) 36
4. P(total shown > 10) 1
12