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**Bellwork You roll a fair die one time, find each probability below.**

P(1 or 2) P(even number) P(2, 3, 4, or 5)

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**Probabilities of compound events**

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Vocabulary A compound event combines two or more events using the word and or the word or.

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**Examples of Compound Events…**

The probability that when you roll a dice, you roll a 1 or a 4. The probability that when you pick a card from a deck, you choose an ace and a red card. The probability that when you reach into a bag of coins, you choose a dime or a penny.

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Let’s start by examining compound events that use the word or, as in “find the probability of A or B”.

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**In probability notation:**

P(A or B) In this case, events A and B are either mutually exclusive events, or overlapping events.

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**Vocabulary Mutually exclusive events have no common outcomes.**

P(A or B) Mutually exclusive events have no common outcomes. Overlapping events have at least one common outcome.

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**Let’s look at an example of a mutually exclusive event…**

In a deck of cards, you choose an Ace or a King. Since King’s cannot also be Aces, the event is mutually exclusive.

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**Suppose we choose one person from this class.**

Which of these events are mutually exclusive??? The person is either male or female. The person is either male or hispanic. The person is either blue eyed or brown eyed or green eyed.

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Your turn… You have 3 minutes to come up with as many examples of mutually exclusive events as you can. Let’s see who can get the most...

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One minute left…

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Time’s Up

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**Formula for finding the Probability of mutually exclusive events:**

If A and B are mutually exclusive events, then P(A or B) = P(A) + P(B)

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**Examples from your worksheet:**

You randomly choose a card from a deck of 52 playing cards. Find the probability that you choose a Queen or a Ace: P(Queen or Ace) = P(Queen) + P(Ace) = 4/52 + 4/52 = 2/13 = or 15.3% chance

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Checkpoint… In a deck of cards, there are 52 cards, 26 are black and 26 are red. There are 4 suits (black clubs and black spades, red diamonds and red hearts). Find the probability that… You choose a queen or an odd number. You choose a club or a red card. You choose a red card or you choose a club or a spade.

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**Overlapping Events These are events that can have common outcomes.**

For example, from a deck of cards, what’s the probability I choose a King or a club card? This is an overlapping event because there is 1 card that is both a King and a club.

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**Examples of overlapping events:**

In a deck of cards, you choose a 3 or a diamond. You roll a dice and you get a number that is even or divisible by 3.

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**Formula for finding the probability of overlapping events:**

If A and B are overlapping events, then P(A or B) = P(A) + P(B) – P(both A, B)

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**So, what is the probability I choose a King or a club card?**

P(King) = 4/52 P(club) = 13/52 P(King and club) = 1/52 (remember, there’s only 1 card that is both a King and a club). P(King or club) = 4/ /52 – 1/52 = 16/52 = 4/13

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Checkpoint Complete the bottom part of your worksheet, a) through d).

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Bellwork In a deck of cards, there are 52 cards, 26 are black and 26 are red. There are 4 suits (clubs, spades, diamonds, and hearts). There are 13 cards in each suit. Tell whether each even below is overlapping or mutually exclusive. Then give the probability of each. 1. You choose a heart or a spade. 2. You choose a 3 or an odd number. 3. You choose a red card or a black card.

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**Yesterday we looked at compound events that had the word or**

Yesterday we looked at compound events that had the word or. Now, let’s look at compound events that use the word and, as in, find the probability of A and B.

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P(A and B) In this case, events A and B are either: independent events, or dependent events.

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Vocabulary Two events are independent events if the occurrence of one event as no effect on the occurrence of the other. Two events are dependent events if the occurrence of one event affects the occurrence of the other.

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**Let’s look at an example of an independent event…**

You draw a name from a hat, replace it, and then draw another name from the hat. You choose one girl from this class, and then you choose one boy from this class.

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**Now let’s look at an example of a dependent event…**

You draw a name from a hat and keep it. Then you draw a second name from the hat. You randomly select one student from this class, remove that student, then randomly select a second student.

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**Checkpoint: Tell whether these events are dependent or independent:**

An aquarium contains 6 male fish and 4 female fish. You randomly select one fish to purchase. Then you select a second fish. An aquarium contains 6 male fish and 4 female fish. You randomly select one fish, but it isn’t the gender you wanted, so you put it back in. Then you select a second fish.

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Your turn… Create an example of a dependent event and an independent event.

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**Formula for finding the Probability of independent events:**

If A and B are independent events, then P(A and B) = P(A) x P(B)

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Let’s try a few… You draw a card from a deck, replace it, then draw another card. What is the probability you pick a queen on the first and a 7 on the second? P(queen and 7) = P(queen) x P(7) = 4/52 x 4/52 = 16/2704 = 1/169

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Let’s try a few… You roll two dice. What is the probability you roll a 5 on the first and a 2 on the second? Write it in probability notation. P(5 and 2) = P(5) * P(2) = 1/6 * 1/6 = 1/36

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Checkpoint Complete the left side of the table on your worksheet.

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**Formula for finding the Probability of dependent events:**

If A and B are dependent events, then P(A and B) = P(A) x P(B given A)

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**Conditional Probability**

Instead of writing P(B given A) we usually write P(B|A). They mean the same thing.

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Let’s try a few… You draw a card from a deck, keep it, then draw another card. What is the probability you pick a queen on the first and a 7 on the second? P(queen and 7) = P(queen) x P(7|queen) = 4/52 x 4/51 = 16/2652 = 4/663

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Let’s try a few… A bag has 5 blue marbles, 8 red marbles and 3 yellow marbles. You choose a marble, do not replace it, then choose a second marble. What is the probability you choose a red marble on both your first and second try? P(red and red) = P(red) x P(red|red) = 8/16 x 7/15 = 56/240 = 7/30

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Checkpoint… Complete the right side of your worksheet.

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Ticket out the door Tell whether each event is mutually exclusive, overlapping, independent, dependent. Then find the probability. 1. You draw a card from a deck, replace it, then draw another card. What is the probability you pick a king on the first and a 4 on the second?

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Ticket out the door Tell whether each event is mutually exclusive, overlapping, independent, dependent. Then find the probability. 2. You draw a card from a deck, do not replace it, then draw another card. What is the probability you pick a king on the first and a 4 on the second?

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Ticket out the door Tell whether each event is mutually exclusive, overlapping, independent, dependent. Then find the probability. 3. You draw a card from a deck. What is the probability you pick a king or a 4?

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Ticket out the door Tell whether each event is mutually exclusive, overlapping, independent, dependent. Then find the probability. 4. You draw a card from a deck. What is the probability you pick a king or a heart?

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