Transcript Probability Test Review

Probability Test Review
(What are your chances of passing?)
See if you know the probability of:
• Rolling a 3 with two dice?
• 2 out of 36 = 1/18th
5.6%
• Rolling a 9 with two dice?
• 4 out of 36 = 1/9th
11.1%
• Rolling “doubles?”
• 6 out of 36 = 1/6th
16.7%
• Rolling a 7 with two dice?
• 6 out of 36 = 1/6th
16.7%
How many ways….
• To roll two dice?
• 6 × 6 = 36 ways
• To roll three dice?
• 6 × 6 × 6 = 216 ways
• To choose 4 people from a group of 10? (order does not matter)
•
10!
4! 10−4 !
= 10C4 = 210 ways
• To arrange 5 people in a line? (order does matter)
• 5! = 5 × 4 × 3 × 2 × 1 = 120 ways
Odds vs. Probability
• The probability of rolling a 7 with two dice is 1/6th . What are the
odds in favor? Odds against?
• Odds in favor are 1 to 5. Odds against are 5 to 1.
• The two numbers in the odds ratio add up to the total number of
ways in which an event can occur. Frequently that ratio can be
simplified. Suppose there are 100 possibilities in a game, and 40 ways
to win, 60 ways to lose. What are the odds in favor of winning?
• 40 to 60 = 4 to 6 = 2 to 3. Odds against are 3 to 2.
• If the probability in favor is 3/7th then what are the odds against?
• 4 to 3 against. (3 ways to “win” 4 ways to lose, lose to win = 4:3)
Pascal’s Triangle
• “zero” row
1
• “1” row
1 1
• “2” row
1 2 1
•
1 3 3 1
•
1 4 6 4
1
•
1 5 10 10 5 1
•
1 6 15 20 15 6 1
• What are the numbers in the “9” row?
•
1 9 36 84 126 126 84 36 9 1
Flipping a Coin or True-False using Pascal’s Triangle
• What’s true about the sum of every row in Pascal’s Triangle?
• They are all powers of 2: 1…2…4…8…16…32…64…128…256…512…
• What’s also true about each number in Pascal’s Triangle?
• Each one is a combination number. For example, if you have 9 items
and choose any 4 at random, that’s 9C4 = 126 ways.
• How many ways could you get (by guessing alone) exactly 7 questions
right on a ten-question true-false test? (See the “10” row)
• 10C7 = 120 ways
• Flip ten coins, exactly 7 come up heads? Same answer: 120 ways
Adding Probabilities
• What about “at least” getting 7 out of 10 right by guessing alone?
• Use the “10” row of Pascal’s Triangle:
• 1 10 45 120 210 252 210 120 45 10 1
• Add the values for 7 out of 10 (120), 8 out of 10 (45), etc.
• 120 + 45 + 10 + 1 = 176 different combinations.
• The probability is 176 out of 1024. Simplify that fraction.
• 11/64 That’s just a bit above 17% (0.171875)
• How about getting at least half right by guessing?
• 638 / 1024 ≈ 62.3 %
Playing Cards
• There are 52 cards in a standard deck of cards. (without jokers)
• Any combination of a “hand” will most likely start with 52.
• How many five-card hands are there?
• Use 52C5
How many possibilities are there?
• 2,598,860 Wow! How many “Spades” hands? (13 cards)
• 52C13 = ??
• 635,013,559,600 Boy! That’s a lot of different combos!