Download Probability

Probability
Objectives
SWBAT:
• Interpret probabilities as ratios, percents, and decimals
• Find and estimate probabilities and make predictions based on theoretical and experimental probabilities
• Model situations involving probabilities with simulations and theoretical models
Feb 15­7:07 PM
Feb 20­12:50 PM
Probability Basics Feb 20­2:20 PM
Probability
Probability: tells how likely an event is to happen; "What's the chance...?"
• Can be expressed as a ratio (fraction), percent, or decimal between 0 and 1 0 <­­­­­­­­­­­­­­­1/4­­­­­­­­­­­­­­­­­1/2­­­­­­­­­­­­­­­­3/4­­­­­­­­­­­­­­­­­­> 1 impossible unlikely equally likely likely certain 0 % 25% 50% 75% 100%
0 .25 .50 .75 1
Complement of an event A: not A Probability of an event and its complement
= 1 or 100%
ex: The weather man states that there is a 40% chance of rain tonight. What is the probability that there will not be rain tonight?
Feb 15­7:07 PM
Theoretical Probability
• Based on mathematical reasoning
• What is “expected” to occur
P (event) = number of successful outcomes
total number of possible outcomes
outcome: possible result
equally likely outcomes: outcomes that have the same probability of occurring ex: roll a die: 1, 2, 3, 4, 5, 6 are equally likely
toss a coin: heads or tails are equally likely
(spinner: must have equal sections)
sample space: a list of all possible outcomes
ex: roll a die (number cube): Sample space S = {1,2,3,4,5,6}
Feb 15­7:07 PM
How many outcomes are possible when you toss a coin?
How many outcomes are possible when you toss 2 coins at the same time?
Feb 10­2:04 PM
How many outcomes are possible if you toss 3 coins at the same time?
Feb 10­1:57 PM
1. Roll a die/number cube:
a. How many possible outcomes are there?
b. Find P (even number)
c. Find P (> 2)
2. Roll 2 dice/number cubes: a. How many possible outcomes are there?
b. Find P (sum is 9)
c. Find P (sum is 12)
d. Find P (sum is 1)
Feb 15­7:07 PM
A bag contains 20 marbles: 6 red, 8 black, 4 blue, 2 yellow
You pick one at random.
Determine the following:
a. P (yellow)
b. P (red or blue)
c. P (not black)
d. P (white)
Feb 10­2:24 PM
P (red) =
P (red) =
P (white) =
P (blue) =
P (not green) =
Feb 10­2:36 PM
P (tails) =
P (heads) =
P (even number) =
P (red) =
P (2) =
P (black) =
Feb 15­7:13 PM
"Theoretical" Probabilities
Examples of simple events: 1. toss a coin à 2 possible outcomes (Heads or Tails)
P (heads) = 1/2 P (tails) = 1/2 2. roll a die or number cubeà 6 possible outcomes (1, 2, 3, 4, 5, 6)
P (4) = 1/6
P (odd number: 1,3,5) = 3/6 = ½
3. pick a marble from a bag: 5 green, 3 blue, 6 red, 1 yellow
(15 possible outcomes)
P (red): 6/15 = 2/5
= 0.4
= 40%
Feb 15­7:07 PM
Experimental Probability
• based on actual experiments & repeated trials
EXP PROB. (event) = number of successes
total number of trials
*** the more trials done, the closer the experimental probability will be to the theoretical probability ***
Feb 15­7:07 PM
For each of the situations below, what is the theoretical probability ? experimental probability? 1. Toss a coin 30 times
P (heads) P (tails)
Heads
Tails
2. Spin a spinner 20 times P (red)
Blue
Green
Red
Yellow
Number 1
Frequency 9
2 3
4
5
6
17 16 10 15 13 3. Roll a number cube 80 times. What is the theoretical probability of a 3? 1?
Find each experimental probability (as a fraction & percent) based on the frequencies shown in the table: P (3) P (1) 4. Roll a number cube 20 times
P (5) Number
1
2
3
4
Frequency
Feb 15­7:07 PM
5
6
Melissa randomly draws a marble from a bag of 72 marbles, then replaces the marble. She does this 30 times and places her results in the frequency table. Based on these results, how many marbles in the bag are white?
white:
Color
Frequency
black
10
white
5
grey
15
5 = x 30
72
actual
results
TOTAL 30
Feb 20­5:53 PM
Julio randomly picked a marble from a bag , then replaced the marble. He did this 25 times and his results are shown in the frequency chart. Based on the table, how many times can Julio expect to draw a red marble in 200 trials?
Color
Frequency
red
16
white
4
blue
5
TOTAL 25
red:
16 = x 200
25
actual
results
Feb 20­5:53 PM