Warm-up

Would it be fair to give a report card grade based on 1 test? or 1 assignment?

Would it be accurate to conclude that a coin will always come up heads after flipping it once?

If 50% of students in a class said they like country music, do you think that means 50% of students in the whole school like country music?

Could you assume that if a person throws a basketball once and makes a basket from half court, then they are a good shooter?

S

Classical and Empirical Probability

Theoretical and Experimental

Classical/Theoretical Probability

Assumes that we are given a situation in which all outcomes are equally likely to occur.

A penny and a quarter are tossed once. If this experiment is repeated

1000 times, what is the expected number of times that the results show

tails for both coins?

If A is the event of getting both tails, we can write

A = {(T, T)} We can also write

P(A) =

€

14

€

141000( ) =250

Suppose that B is the event of getting one tail

and one head.

Then B = {(H, T), (T, H)} and P(B) =

€

24=12

A nickel, a quarter, and a silver dollar are tossed once. If C represents the

event of getting exactly one tail and two heads, write C as a set of elements.

C = {(T, H, H), (H, T, H), (H, H, T)}.

Suppose the experiment from the previous slide is repeated 736 times. What is the expected number of times

that the results show exactly one tail and two heads?

P(C) = , then we can expect to get exactly one tail and two heads is

€

38

€

38

⎛

⎝ ⎜

⎞

⎠ ⎟ 736( ) =276

An ordinary die is rolled once. Event D is defined as the set of outcomes in

which a prime number is shown. What is the value of P(D)?

D= {2, 3, 5}

P(D) =

€

36=12

Suppose this experiment is

repeated 246 times.

€

12

⎛

⎝ ⎜

⎞

⎠ ⎟ 246( ) =123

In drawing two cards from a deck of cards, one card at a time, with the replacement of the first

card prior to drawing the second card. Let F represent the event of drawing 2 red jacks. What

is the value of P(F)?

F = {(JD, JD), (JD, JH), (JH, JH), (JH, JD)}

So P(F) =

€

42704

=1676

€

1676

⎛

⎝ ⎜

⎞

⎠ ⎟ 5408( ) =8

Suppose this experiment is repeated 5408

times.

Empirical/Experimental Probability

Assumes that the outcomes are not equally likely; rather, their associated probabilities are calculated based on observations or historical data.

A particular die is “weighted”, which means that it tends to land on 4, 5, or 6 more often than it does on 1, 2, or 3. In rolling this die 1800 times, the results of the frequency of

each outcome are as follows: Outcome Frequency

1 120

2 140

3 250

4 450

5 300

6 540

Based on this chart, what is the probability that in rolling this die, it will land on a 3 or a 4?

€

250 +450 =700

The required probability is7001800

This “weighted” dime is tossed 3 times, and the experiment is repeated 4000

times. Here are the results:

Outcome

Frequency

HHH 2050HHT 510HTH 430HTT 240THH 500THT 100TTH 120TTT 50

Based on this chart, what is the probability that when this dime is tossed three time, the result will be exactly two tails?

€

450 +100 +120 =460

The required probability is4604000

=23200

Math Flash!!!

When computing a probability value involving coins or dice, use the classical probability approach, unless you are given a chart of observed frequencies from which the probability is to be calculated.

Theoretical or Experimental

1. Maria flipped a coin and got 6 heads out of 10 flips.

2. Carlos said the chances of rain today are 30%.

3. James said he has a 70% chance of making a free throw because yesterday he made 7 out of 10.

4. 6 students out of 18 students in one classroom caught a cold, so the nurse said about 33% of the students in school would catch the same cold.

5. Julia placed an eraser under one of four cups and told Patrick he had a 25% chance of finding the correct cup.

experimental

theoretical

theoretical

experimental

experimental