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Notes on PROBABILITY
What is Probability? Probability is a number from 0
to 1 that tells you how likely something is to happen.
Probability can be either theoretical or experimental.
# of ways can occur( )
total # of possible outcomes
EP E
# sides that are heads 1( ) 0.5
total number of sides 2P E
If you flip a coin, the probability it will land
on heads is ½. P(heads)
A new baby will be
boy girl
If you toss a coin it will land
Heads up Tails up
The probability of the baby being a boy is…
impossible certain50 - 50
0 1½
50 - 50 1 in 2 ½
The probability of a coin landing heads up is also 50 – 50, 1 in 2, or ½.
The probability number line
Impossible events have a probability of 0
impossible certain50 - 50
0 1½
Events that are certain are given a probability of 1
Like all number lines we can mark other points on it.
0 1½¼ ¾
If I pick a card at random from a stack of 4 cards that are numbered 1, 2, 3, 4, the probability that it will be a three is…?
Can you work out why?
0 1½¼ ¾
If I pick a card at random from that same pack of cards, the probability that it will be an even number is…?
Can you work out why?
ProbabilityTHEORETICAL
Theoretical probability can be found without doing an experiment. It is what is supposed to happen.
EXPERIMENTAL
Experimental probability is found by repeating an experiment and observing the outcomes.
THEORETICAL PROBABILITY
I have a quarter.My quarter has a heads side and a tails side.
Since my quarter has only 2 sides, there are only 2 possible outcomes when I flip it. It will either land on heads, or tails.
HEADS
TAILS
THEORETICAL PROBABILITY
When I flip my coin, the probability that my coin will land on heads is 1 in 2.
What is the probability that my coin will land on tails?
HEADS
TAILS
Theoretical Probability
Right!!! There is a 1 in 2 probability that my coin will land on tails!!!
HEADS
TAILS
A probability of 1 in 2 can be written in three ways:
•As a fraction: ½•As a decimal: .50
•As a percent: 50%
Theoretical probability
When I spin this spinner, I have a 1 in 4 chance of landing on the section with the red A in it.
A
A
A
A
A 1 in 4 chance can be written 3 ways:
As a fraction: ¼ As a decimal: .25 As a percent: 25%
A
A
A
A
Theoretical Probability
Other examples of theoretical probability are found in determining the probability of drawing a certain card from a standard deck of cards.
A standard deck has four suits: spades (), hearts (), diamonds (), and clubs (). It has thirteen cards in each suit: ace, 2, 3, 4, 5, 6, 7, 8, 9, 10, jack, queen, and king. Each of these cards is equally likely to be drawn.
The probability of drawing a king is given by:
# of kings in a deck 4 1( )
total # of cards in a deck 52 13P king
The probability of drawing a heart is given by: # of hearts in a deck 13 1( )
total # of cards in a deck 52 4P heart
The probability of drawing a face card (jack, queen, king) is given by:
# of face cards in a deck 12 3( )
total # of cards in a deck 52 13P face card
Dice (singular is die) are cubes that have spots on each side. The spots are usually numbered from 1 to 6. When a fair die is tossed, each side has an equally likely chance of ending up on top. The probability of tossing a die and having a 4 end up on top (this is called rolling a 4) is given by:
# of faces with 4 1(4)
total # of faces 6P
The probability of tossing a die and rolling a 7 is given by:
The probability of tossing a die and rolling a number less than 7 is given by:
# faces with 7 0(7) 0
total # of faces 6P
# faces with less than 7 6(less than 7) 1
total # of faces 6P
These examples lead to four rules or facts about probability:
1. The probability of an event that cannot occur is 0.2. The probability of an event that must occur is 1.3. Every probability is a number between 0 and 1 inclusive.4. The sum of the probabilities of all possible outcomes of an experiment is 1.
I am going to take 1 marble from the bag. What is the probability that I will pick
out a red marble?
Theoretical Probability
I have three marbles in a bag.
1 marble is red.
1 marble is blue.
1 marble is green.
Theoretical Probability Since there are three
marbles and only one is red, I have a 1 in 3 chance of picking out a red marble.
I can write this in three ways:
As a fraction: 1/3 As a decimal: .33 As a percent: 33%
Experimental Probability
Experimental probability is
found by repeating an
experiment and observing the
outcomes.
Experimental Probability Remember the bag of marbles? The bag has only 1 red, 1 green, and
1 blue marble in it. There are a total of 3 marbles in the
bag. Theoretical Probability says there is a
1 in 3 chance of selecting a red, a green or a blue marble.
Experimental Probability
Draw 1 marble from the bag.
Marble number red blue green
1 123456
It is a red marble.
Record the outcome on the tally sheet
Experimental Probability Put the red marble back in the bag and
draw again. This time your drew a green marble. Record this outcome on the tally sheet.
Marble number red blue green
1 12 134
Experimental Probability Place the green marble back in the bag. Continue drawing marbles and recording
outcomes until you have drawn 6 times. (remember to place each marble back in the bag before drawing again.)
blue, green, red, green
Experimental Probability After 6 draws your
chart will look similar to this.
Look at the red column. Of our 6 draws, we
selected a red marble 2 times.
Marble number red blue green
1 12 13 14 15 16 1
Total 2 1 3
Experimental Probability The experimental
probability of drawing a red marble was 2 in 6.
This can be expressed as a fraction: 2/6 or 1/3 a decimal : .33 or a percentage: 33%
Marble number red blue green
1 12 13 14 15 16 1
Total 2 1 3
Experimental Probability Notice the
Experimental Probability of drawing a red, blue or green marble.
Marble number red blue green
1 12 13 14 15 16 1
Total 2 1 3
Exp. Prob.
2/6 or 1/3 1/6
3/6 or 1/2
Comparing Experimental and Theoretical Probability
Look at the chart at the right.
Is the experimental probability always the same as the theoretical probability?
red blue greenExp. Prob. 1/3 1/6 1/2Theo. Prob. 1/3 1/3 1/3
Comparing Experimental and Theoretical Probability
In this experiment, the experimental and theoretical probabilities of selecting a red marble are equal.
red blue greenExp. Prob. 1/3 1/6 1/2Theo. Prob. 1/3 1/3 1/3
Comparing Experimental and Theoretical Probability
The experimental probability of selecting a blue marble is less than the theoretical probability.
The experimental probability of selecting a green marble is greater than the theoretical probability.
red blue greenExp. Prob. 1/3 1/6 1/2Theo. Prob. 1/3 1/3 1/3
The more you do an experiment, the closer you
get to the theoretical probability.
Probability Review
Theoretical (what is supposed to happen, can be found without doing an experiment)
Experimental (can be found by repeating an experiment and recording outcomes.)
There are 2 types of probability:
Probability is a number from 0 to 1 that tells you how likely something is to happen.
Probability Review
Probability can be expressed as a fraction, a decimal or a percentage.
Probability fraction decimal percent1 in 2 1/2 0.50 50%
1 in 3 1/3 0.33 33%
1 in 4 1/4 0.25 25%
