Download possible numbers total possible numbers even . . . . 2 1 6 3 =

MA.7.p.7.1 Determine the outcome of an experiment and predict which events are likely or unlikely, and if the experiment is fair or
unfair.
MA.7.P.7.2 Determine, compare, and make predictions based on experimental or theoretical probability of independent or dependent
events.
What is the probability of rolling an even number on a number cube marked with 1, 2, 3, 4, 5, and 6 on its faces?
P(even number) =
even.numbers. possible
3 1
=

6 2
total .numbers. possible
The probability of rolling an even number is ½, 0.5, or 50%.
Try these:
1. The probability of rolling an odd number
2. The probability of rolling a 5 or 6
3. The probability of rolling a prime number
4. The table shows the numbers of brass instrument players in the New York Philharmonic.
Suppose one brass instrument player is randomly selected to be a featured performer.
Find the probability of each event. Write as a fraction in simplest form.
P(trumpet)
P(brass)
P(flute)
P(horn or tuba)
The probability that an event will happen can be any number from 0 to 1, including 0 and 1, as shown on the number line below.
Notice that the probabilities can be written as fractions, decimals, or percents.
Try these:
1. Put the following events in order based on the number line above:
P(the sun will rise tomorrow)
P(heads comes up on the flip of a coin)
P(not pulling a clubs from a deck of cards)
P(it will snow in Orlando on the last day of school)
P(your birthday being on the weekend)
2. Mrs. Davis is teaching her class about probability. She prepared the set of golf balls listed below.




6 red golf balls, each labeled with a different number from 1 to 6
7 green golf balls, each labeled with a different number from 1 to 7
8 blue golf balls, each labeled with a different number from 1 to 8
9 yellow golf balls, each labeled with a different number from 1 to 9
Mrs. Davis put all the golf balls into a sack and mixed them up. Nancy will be the first student to select a golf ball from the sack
without looking. Which of the following outcomes is most likely to occur?
A. Nancy will select a yellow golf ball.
B. Nancy will select a golf ball that is not blue.
C. Nancy will select a golf ball with the number 6 on it.
D. Nancy will select a golf ball with a number on it that is not 1.
Mathematically speaking, a two-player game is fair if each player has an equal chance of winning. A game is unfair if there is not such
a chance.
Try this:
Play this game with a partner. Directions:
• Choose a partner from your group
• Decide who is "odd" and who is "even"
• Roll a pair of dice. Find the product (by multiplying) of the two dice
• Determine who gets the point. Put a tally mark in the person's box.
• The first person up to 10 points wins the game
• Play at least 3 games
1. Do the “even person” and the “odd person” have an equal chance of winning the game? Explain why you think this.
Theoretical probability is based on what should happen when conducting a probability experiment. Experimental probability is based
on what actually happens. The theoretical probability and experimental probability may or may not be the same. As the number of
times an experiment is conducted increases, the theoretical probability and experimental probability should become closer in value.
For example, when flipping a coin, the probability of getting a head is ½. Try flipping a coin the amount of times shown and tally your
results. What was the experimental outcome each time? How close was each to ½?
4 times
Theoretical Probability: How
many heads would be ½?
10 times
Theoretical Probability: How
many heads would be ½?
20 times
Theoretical Probability: How
many heads would be ½?
50 times
Theoretical Probability: How
many heads would be ½?
Experimental Probability:
# of heads:
# of tails:
Experimental Probability:
# of heads:
# of tails:
Experimental Probability:
# of heads:
# of tails:
Experimental Probability:
# of heads:
# of tails:
# of .heads
:
total . flips
# of .heads
:
total . flips
# of .heads
:
total . flips
# of .heads
:
total . flips
Independent Events
Experiment 1: A dresser drawer contains one pair of socks with each of the following colors: blue, brown, red, white and black. Each
pair is folded together in a matching set. You reach into the sock drawer and choose a pair of socks without looking. You replace this
pair and then choose another pair of socks. What is the probability that you will choose the red pair of socks both times?
There are a couple of things to note about this experiment. Choosing a pairs of socks from the drawer, replacing it, and then choosing
a pair again from the same drawer is a compound event (a compound event consists of two or more simple events.) Since the first pair
was replaced, choosing a red pair on the first try has no effect on the probability of choosing a red pair on the second try. Therefore,
these events are independent.
Definition: Two events, A and B, are independent if the fact that A occurs does not affect the probability of B occurring.
Some other examples of independent events are:
 Landing on heads after tossing a coin AND rolling a 5 on a single 6-sided die.
 Choosing a marble from a jar AND landing on heads after tossing a coin.
 Choosing a 3 from a deck of cards, replacing it, AND then choosing an ace as the second card.
 Rolling a 4 on a single 6-sided die, AND then rolling a 1 on a second roll of the die.
To find the probability of two independent events that occur in sequence, find the probability of each event occurring separately, and
then multiply the probabilities. This multiplication rule is defined symbolically below. Note that multiplication is represented by
AND.
Multiplication Rule 1: When two events, A and B, are independent, the probability of both occurring is:
P(A and B) = P(A) · P(B)
Experiment 1: A dresser drawer contains one pair of socks with each of the following colors: blue, brown, red, white and black. Each
pair is folded together in a matching set. You reach into the sock drawer and choose a pair of socks without looking. You replace this
pair and then choose another pair of socks. What is the probability that you will choose the red pair of socks both times?
Probabilities:
1
P(red)
=
5
P(red and red) = P(red) · P(red)
1
1
=
·
5
5
1
=
25
Experiment 2: A coin is tossed and a single 6-sided die is rolled. Find the probability of landing on the head side of the coin and
rolling a 3 on the die.
Probabilities:
1
P(head)
=
2
1
P(3)
=
6
P(head and 3) = P(head) · P(3)
1
1
=
·
2
6
1
=
12
Try these:
1. Spin a spinner numbered 1 to 7, and toss a coin. What is the probability of getting an odd number on the spinner and a tail on
the coin?
2. A jar contains 6 red balls, 3 green balls, 5 white balls and 7 yellow balls. Two balls are chosen from the jar, with
replacement. What is the probability that both balls chosen are green?
3. Four cards are chosen from a standard deck of 52 playing cards with replacement. What is the probability of choosing 4
hearts in a row?
Dependent Events
Experiment 1: A card is chosen at random from a standard deck of 52 playing cards. Without replacing it, a second card is chosen.
What is the probability that the first card chosen is a queen and the second card chosen is a jack?
Analysis: The probability that the first card is a queen is 4 out of 52. However, if the first card is not replaced, then the
second card is chosen from only 51 cards. Accordingly, the probability that the second card is a jack given that the first card
is a queen is 4 out of 51.
Conclusion: The outcome of choosing the first card has affected the outcome of choosing the second card, making these
events dependent.
Definition: Two events are dependent if the outcome or occurrence of the first affects the outcome or occurrence of the
second so that the probability is changed.
Now that we have accounted for the fact that there is no replacement, we can find the probability of the dependent events in
Experiment 1 by multiplying the probabilities of each event.
Experiment 1: A card is chosen at random from a standard deck of 52 playing cards. Without replacing it, a second card is
chosen. What is the probability that the first card chosen is a queen and the second card chosen is a jack?
Probabilities:
4
52
4
P(jack on 2nd pick given queen on 1st pick) =
51
4
4
16
4
P(queen and jack)
=
·
=
=
52 51 2652 663
P(queen on first pick)
=
Try these:
1. Mr. Parietti needs two students to help him with a science demonstration for his class of 18 girls and 12 boys. He randomly
chooses one student who comes to the front of the room. He then chooses a second student from those still seated. What is
the probability that both students chosen are girls?
2. On a math test, 5 out of 20 students got an A. If three students are chosen at random without replacement, what is the
probability that all three got an A on the test?
3. A school survey found that 7 out of 30 students walk to school. If four students are selected at random without replacement,
what is the probability that all four walk to school?
Try these:
1. A discount supermarket has found that 60% of its customers spend more than $75 each visit. What is the probability that the
next two customers will each spend more than $75?
2.
Donoma had 8 dimes and 6 pennies in her pocket. If she took out 1 coin and then a second coin without replacing the first,
what is the probability
3.
Mr. Fernandez is holing four straws of different lengths. He has asked four students to each randomly pick a straw to see who
goes first in a game. Milo picks first, gets the second longest straw, and keeps it. What is the probability that Felipe will get
the longest straw if he picks second?
4.
A jar of beads contains 6 aqua beads and 4 black beads. If two beads are selected at random, with replacement, what is the
probability that both beads will be aqua?
5.
What is the probability of tossing heads on a coin and rolling an odd number on a six-sided number cube?
6.
Suppose Jay spins the spinner once. What is the likelihood of him spinning an A and a 2
A.
B. equally likely
C. likely
D. unlikely
E. Not enough information to determine
7. Joaquin has the set of cards shown below.
Joaquin will shuffle the cards, select one without looking, record the number of the card, and return the card to the set. He
will repeat this process 50 times. Which is closest to the number of times he should expect to select a card with a number
greater than 5?
A. 6
B. 10
C. 19
D. 25
8. A bowl of candies contains 6 candies with nuts and 4 candies without nuts. A piece of candy is selected at random and eaten.
Then a second piece of candy is selected at random. What is the probability that the first piece will have nuts and the second
piece will not have nuts?