2. Warm Up: Quick Write Which is more likely, flipping exactly 3 heads in 10 coin flips or flipping exactly 4 heads in 5 coin flips ?

3. Sample Space  A set or list of possible outcomes. Examples: 1. When a coin is flipped, there are two possible outcomes, a head(H) or tail(T). So: S = { H,T} 2. When a coin is flipped twice, there are four possible outcomes: S = { HH, HT,TH,TT)

4. 1. When rolling a dice, the list of possible outcomes are 1, 2, 3, 4,5, 6. So, the sample space is: S= { 1,2,3,4,5,6}

5. 1 2 3 4 6 1 (1,1) (1,2) (1,3) (1,4) (1,6) 2 (2,1) (2,2) (2,3) (2,4) (2,6) 3 (3,1) (3,2) (3,3) (3,4) (3,6) 4 (4,1) (4,2) (4,3) (4,4) (4,6) 5 (5,1) (5,2) (5,3) (5,4) (5,6) 6 (6,1) (6,2) (6,3) (6,4) (6,6) 1. When rolling a dice twice, the list of possible outcomes are 1, 2, 3, 4,5, 6. So, the sample space is: S = { (1,1),(1,2),…….,(6,6)}

6. What about a deck of cards?

7. Event  One or more outcome. It is a subset of the sample space.  The number of outcomes in an event is denoted by { A}  Examples:  1. When you toss a coin once and the head comes out, then the head(H) is an event. The number of outcomes of a head is {1}.

8.  2. Getting a Tail when tossing a coin is an event  3. Rolling a "5“ in a dice is an event.  4. Rolling an "even number" (2, 4 or 6) is an event  5. Rolling a prime number a dice is an event  6. Getting a spade in a deck of cards is an event  What could be an event when you flip a coin twice?

9. Equally Likely  If each outcome has the same chance of occurring, then the events are equally likely.  Examples:  In flipping a coin, the event of a head(H) or tail ( T) is equally likely.  In rolling a cube, each of the faces are equally likely.  In drawing a card from a deck of cards, each of the cards is equally likely.

10. IIn the spinner below: 1.Are the outcomes blue and yellow equally likely? 2.Are the outcomes red and blue equally likely? 3.Are the outcomes red and yellow equally likely?

11. Simple events  Has only one or single outcome in the sample space.  Examples:  A single face when a die is rolled is a simple event like getting 5 or 6 or 3 or 2 etc…  Getting a head when you flip a coin is simple event  Getting two heads (HH ) in flipping a coin twice is a simple event. This is because there is only one and only one outcome of a HH in the sample space of flipping 2 coins: S= { HH,HH,TH, TT}  Is the event of getting at least one head a simple event?

12. Compound Events  A combination of two or more than two simple events. It can also be defined as an event that occurs more than once in the sample space. Examples:  Getting an even number when a dice is rolled  Getting at least a head in tossing 2 coins ( HH,HT,TH)  An event that consists of the sum of two dice is ‘’5’’(it consists of four outcomes i.e., (1, 4), (2, 3), (3, 2), (4, 1) )  Is the event of drawing a King in a deck of cards a compound event?

13. Mutually Exclusive Events  Mutually Exclusive means we can't get both events at the same time. It is either one or the other, but not both. Examples:  Turning left or right are Mutually Exclusive (you can't do both at the same time)  Heads and Tails are Mutually Exclusive  Kings and Aces are Mutually Exclusive

14. Are Kings and Hearts Mutually Exclusive?

15. Independent Events  A event not affected by any other events.  Two events, A and B, are independent if the fact that A occurs does not affect the probability of B occurring. Example: You toss a coin three times and it comes up "Heads" each time... what is the chance that the next toss will also be a "Head"?  The chance is simply 1/2, or 50%, just like ANY OTHER toss of the coin.  What it did in the past will not affect the current toss!  Choosing a marble from a jar AND landing on heads after tossing a coin.  Choosing a 3 from a deck of cards, replacing it, AND then choosing an ace as the second card.  Rolling a 4 on a single 6-sided die, AND then rolling a 1 on a second roll of the die.

16. Dependent Events  some events can be "dependent"... which means they can be affected by previous events  Example: Drawing 2 Cards from a Deck  After taking one card from the deck there are less cards available, so the probabilities change!   Let's look at the chances of getting a King.  For the 1st card the chance of drawing a King is 4 out of 52  But for the 2nd card:  If the 1st card was a King, then the 2nd card is less likely to be a King, as only 3 of the 51 cards left are Kings.  If the 1st card was not a King, then the 2nd card is slightly more likely to be a King, as 4 of the 51 cards left are King.

17. Remember!  Drawing a card from a deck of cards with replacement is an independent event  Drawing a card from a deck of cards without replacement is a dependent event

18. Stop Point: Review Which events are equally likely? 1. A and B are playing tennis. The event that A or B will win the match. 2. Dennis has been playing professional football for ten years and invited Ken to play with him one on one. Ken agreed. If ever, this is the first time Ken will play football, The event that Dennis or Ken will win the match.

19. Check Point  Tell which are dependent or independent events!  1. Roll a die; toss a coin.  2. Take a marble out of a bag;  take a second marble out of a bag.  3. Choose a person from a group of 50 persons.  Choose another person from the same group.  4. Draw a card from a deck and put it back.  Draw another card again.

20. Probability  Measures how likely it is for an event to occur.  It is the number of times an event A will happen divided by the total number of possible outcomes as shown in the sample space.  It is expressed either as a fraction, a percentage or decimal. The range of values of a probability is between 0 to 1.  P(A) = number of outcomes of A  _____________________  total number of outcomes Examples:  The probability in getting exactly two heads in two coin tosses is 1/4.  The probability of rolling an even number in a dice is 3/6.  The probability in drawing a King in a deck of cards is 4/52.

21. Suppose we have a deck of cards…  What is :  P( ♥ )?  P(Jack)?  P(5 ♣ ) ?  P(3 ♥ then Q ♠ from the same deck)?   P(A ♦ from first deck & a ♠ from the second deck)?   P(8 & K)?   P(7 or Red)?

22. Probability of Independent Events  If two events A and B are independent, the probability that A and B will happen is given by:  P(A and B) = P(A) · P(B)

23. Example 1  What is the probability of getting 2 heads in two tosses of a coin?  A = event of a head in first toss  B = event of a head in second toss  P(A and B) = P(A) x P(B)  = ½ x ½  = ¼ 

24. Example 2 

25. Example  A coin is tossed and a single 6-sided die is rolled. Find the probability of landing on the head side of the coin and rolling a 3 on the die.  A = head B = rolling a 3 on the die  P(A and B) = P(A) x P(B)  = ½ x 1/6  = 1/12