Download Probability

What is Probability?
• The study of probability helps us figure out
the likelihood of something happening.
• In math we call this “something
happening” or an “event”
• Probability is a way of expressing what the
chances are that an event will occur
The term probability is common
• Weather forecast - 30 % chance of rain
• Gambling - two dice will produce a sum of 7 is 1/6
Things to know:
1st thing:
What the probability statements mean
2nd thing:
Then we consider how we determine the numerical value
for that probability
Probability Problems
• Problem: A spinner has 4 equal sectors
colored yellow, blue, green, and red. What
are the chances of landing on yellow after
spinning the spinner?
Solution: The chances of landing on
yellow are 1 in 4, or one fourth.
Probability Problems
• Very important to identify all of the different
outcomes that could occur
• Outcomes = the different results that could
happen in the event or the different possibilities
• To find a basic probability with all outcomes
equally likely, we use a fraction:
Probability =
number of favorable outcomes
total number of possible outcomes
What is a favorable outcome?
• Example: A spinner has 4 equal sectors
colored yellow, blue, green and red. Want
to find the probability of the spinner
landing on yellow.
• Favorable outcome =landing on yellow
section
The probability of landing on yellow
is 1 in 4 or 1/4
Experiment: A spinner has 4 equal sectors colored
yellow, blue, green and red. After spinning the spinner,
what is the probability of landing on each color?
Possible Outcomes: The possible outcomes of this
experiment are yellow, blue, green, and red.
Probabilities:
P(yellow) = number of ways to land on yellow = 1
total number of colors
4
P(blue) = number of ways to land on blue = 1
total number of colors
4
P(green) = number of ways to land on green = 1
total number of colors
4
P(red) = number of ways to land on red = 1
total number of colors
4
• Example: Number of ways 2
dice would produce a sum of 7
Die
1
Die
2
Die
1
Die
2
Die
1
Die
2
Die
1
Die
2
Die
1
Die
2
Die
1
Die
2
1
1
2
1
3
1
4
1
5
1
6
1
1
2
2
2
3
2
4
2
5
2
6
2
1
3
2
3
3
3
4
3
5
3
6
3
1
4
2
4
3
4
4
4
5
4
6
4
1
5
2
5
3
5
4
5
5
5
6
5
1
6
2
6
3
6
4
6
5
6
6
6
• 36 different combinations
• Shows all the different ways the dice can
land
• Shows all the ways you could roll a seven
• 6 of the 36 produce the sum of 7
• Must assume that each of the 36
combinations are equally likely to occur
• So there is a 1/6 chance that you the sum
of the dice will be 7
What is the total number of
possible outcomes?
• Is called a sample space
• Sample space is a set consisting of all the
possible outcomes of an event.
The number of different ways you can
choose something from the sample space
is the total number of possible outcomes.
• Example: Jar with 4 red marbles and 6
blue marbles
Want to find the probability of drawing a
red marble at random.
Favorable outcome = drawing a red marble
So we have 4 red marbles and 6 blue
marbles in our jar
Sample space = all ten marbles because
we are likely to draw any one of them
Favorable outcomes = # of red marbles = 4
Possible outcomes = total # of marbles = 10
(sample space)
4/10 reduces to 2/5
Probability of drawing a red marble where all
outcomes are equally likely is 2/5
Ways to Express Probability
As a fraction~ 4/10 = 2/5
As a decimal ~ 4/10 = .4
As a percent ~ 4/10 = 40/100 = 40%
Unlikely events have a probability near zero
Likely events have probabilities near 1
• Probability is a fraction of the sample
space
• The sum of the probabilities of all the
possible outcomes equals 1
• The probability of the occurrence of an
event is always 1 minus the probability
that it doesn’t occur
P(yellow) = number of ways to land on yellow = 1
total number of colors
4
P(blue) = number of ways to land on blue = 1
total number of colors
4
P(green) = number of ways to land on green = 1
total number of colors
4
P(red) = number of ways to land on red = 1
total number of colors
•
•
•
•
4
Probability = ¼
Sample space = 4 colors
Sum = ¼ + ¼ + ¼ + ¼ = 1
Probability of NOT landing on
yellow = 1-1/4 = 3/4
Spinner
problem
Marble Problem
• Probability of picking a red marble was
4/10 or 2/5
• Sample space = 10 marbles in the jar
• So the probability of not picking a red
marble
1 = 10/10
10/10 -4/10 = 6/10 or 3/5
(this is also the probability of picking a
blue marble)
• We only have 2 events with our red and
blue marbles
• Either we pick a red marble or a blue
marble
• If you don’t do the first, then you must do
the second
• So the probability of picking a red marble
plus the probability of picking a blue
marble equals 1 or 100%
Sum = 4/10 + 6/10 = 10/10 = 1
Situation with 2 events
• You draw 1 marble from the 10
• Then I draw another marble from the nine
that remain
What is the probability that I will draw a blue
one first?
What is the probability that you will draw a
red one second?
• Your probability of drawing a blue one is
6/10
• After you draw there are only 9 marbles
left and 4 of those are red, so the
probability that I will draw a red one is 4/9
When there are 2 events, the second
outcome is dependent on the first.
Classroom Exercise/Homework
What can you learn
from the chart?
%’s of
colors in
different
kinds of
M & M’s
1/2007