Bell Work

• Today’s problems are exploratory. I want to see how you react to them.

• Think about these problems and the situations. We’ll see if you change the way you think about them in the coming week!

Agenda

• Bell Work• Papers IN!- Anyone Want to Share?• Laws of Probability• Probability Problems• Homework 10- Introductory Probability

Problems

OBJECTIVES

• I WILL BE ABLE TO:– Find the probability for independent events- Using the Laws of Probability

EQ:/ What are the rules of the game of probability?

M&M TIME!YAY!

• Supplies:– 1 bag of 30 M&Ms– M&M worksheet

• Directions:– Record the number of red, orange, yellow, blue,

green, brown M&Ms you have

M&M’s

• What is our sample space?– Sample Space: the set of all possible outcomes.– To save time, we’ll use the data of Mr. Benzel’s M

and Ms.

M&Ms—Put ‘em back in their bag!

• PROBABILITY RULE #1: Any probability is a number between 0 and 1.

– What is the probability of pulling out a red M&M?– What is the probability of pulling out a blue

M&M?

M&Ms

• PROBABILITY RULE #2: All possible outcomes together must have a probability of 1.

• Prove It!: Is this true for our M&M’s?

M&M’s

• PROBABILIITY RULE #3: The probability that an event does not occur is 1 minus the probability that the event does occur. – We call the probability that an event will no occur the complement

• What is the probability of NOT pulling out a yellow M&M?

M&M’s

• PROBABILITY RULE #4: If two events have no outcomes in common, the probability that one or the other occurs is the sum of their individual probabilities.

• What is the probability of pulling out a red or a green M&M?

• What is the probability of pulling out an orange or a brown M&M?

M&Ms—Replacement

• Randomly pull out 2 M&Ms from your bag.

• Now, what is the probability of pulling out a red & a green M&M?

• What is the probability of pulling out a blue and a yellow M&M?

REPLACEMENT

• When we are doing probability experiments, we can choose to replace or not replace after each round.

– We did not use replacement in our last M&M example…so we had to change our numbers.

INTERESTING PROBABILITIES

• Life Insurance – We cannot predict whether a particular person

will die this year– BUT…The National Center for Health and Statistics

says that the proportion of men aged 20 to 24 years of age who die in any year is 0.0015. For women, the probability is 0.0005.

– Do you think the insurance company charges more for the man or for the woman?

MYTHS!

• SHORT-RUN REGULARITY: We want to think that events are predictable in the short run, but they aren’t!

MYTHS!

• SURPRISE MEETINGS: When we run into an old friend randomly, we often think it was “meant to be.” The likelihood of running into a particular friend is low, but most people have 1500 acquaintances, so the likelihood or running into someone is rather high.

MYTHS!

• LAW OF AVERAGES: If you toss a coin six times and get TTTTTT, the next toss must be more likely to be a head. THIS IS NOT TRUE!

• Coins and dice have no memory.

• What about the sex of babies?

ROLLING TWO DICE

• Imagine rolling two fair, six-sided dice, one red and one green.

• How should we assign probabilities to the outcomes?

• What is the probability that the sum of the two dice is 5?

• What is the probability that the sum of the two dice is not 5?

THINK AGAIN…

• What is the probability of rolling two dice with a sum of 8?

THINK AGAIN…

The table below shows the proportion of women aged 25 to 29 who have each marital status

What is the probability that a woman is not married?

Martial Status

Never married

Married Widowed Divorced

Probability 0.506 0.452 0.002 0.04

BRAIN WARM-UP

• If you toss a fair coin three times, what’s the probability of getting two heads and one tail?– What is the sample space?

MORE COINS…

• Consider these events:– A=getting 2 heads and one tail.– B=getting three heads– C=getting more heads than tails

• What is the relationship between P(A), P(B), and P(C)?

PAIR-A-DICE—YOU TRY!

• Imagine rolling two fair, six-sided dice—one red and one green. Find the probability of each of the following events:– D=doubles (the same number on both dice)– M=sum of the spots showing on the two dice is 10

or less– R=red die has higher number of spots than green

die

MORE THAN ONE EVENT!

• Let’s say we have a standard deck of playing cards.– 52 cards– 4 suits: clubs, diamonds, hearts, spades– Each suit has: 2, 3, 4, 5, 6, 7, 8, 9, 10, jack, queen,

king, ace

MORE THAN ONE EVENT!

• We shuffle thoroughly and deal one card.

• Let A=getting a face card (jack, queen, or king)• Let B=getting a heart

HEART NOT A HEART

FACE CARD

NOT A FACE CARD

MORE THAN ONE EVENT!

• What is the probability of getting a face card and getting a heart?

• What is P(A and B)=P(face card & heart)

MORE THAN ONE EVENT!

• What is the probability of getting a face card or getting a heart?

• What is P(A or B)=P(face card or heart)

GENERAL ADDITION RULE

• If A and B are any two events resulting from some chance process, then:

P(A or B)=P(A) + P(B) – P(A and B)

Give It a Try!

• If we are rolling two fair, 6-sided dice, what is the probability of rolling doubles or a sum of 6?

EXAMPLE: WHO HAS PIERCED EARS?Pierced Ears?

Gender Yes No TotalMaleFemale

1984

714

9088

Total 103 75 178

•If we randomly select a student from the class, what’s the probability that the student has pierced ears? •If we randomly select a student from the class, what’s the probability that we choose a male with pierced ears? •If we randomly select a student from the class, what’s the probability that we choose someone with pierced ears or a male?

Homework 10

• Introductory Probability Problems• Please try your best on them. They don’t need

to be 100 percent. Think of these as an exploration right now.