Lesson 1 Quiz
(d) A coin is tossed 𝑛 times. Find a formula for the probability that it lands heads on all 𝑛 tosses.
(independent) 1/2 x 1/2 x ... = (1/2)^n
(c) A coin is tossed three times. What is the probability that it lands heads, then tails, then heads?
(independent) 1/2 x 1/2 x 1/2 = 1/8
(e) Roll a die and draw a card from a well-shuffled deck. What is the probability you roll a 3 and draw the queen of diamonds?
(independent) 1/6 x 1/52
(b) A die is rolled twice. What is the chance it lands on five, then on two?
(independent) 1/6 x 1/6 = 1/36
(c) Roll a die. What is the chance of rolling an odd number?
1 or 3 or 5 chance = 1/6 + 1/6 + 1/6 = 1/2
(a) A coin is tossed twice. What is the probability that it lands heads both times?
1/2 x 1/2 = 1/4
As a warm-up, in a room full of 2 people, what is the chance two people have the same birthday?
1/365
(d) An urn has 2 green, 3 blue, and 4 red marbles. If you draw a marble at random, what is the chance it is green or red?
2/9 + 4/9 = 6/9
How about a room full of 3 people?
365/365 x 354/365 x 363/365
(a) Draw four cards from a well-shuffled deck. Find the probability that all four are aces.
4/52 x 3/51 x 2/50 x 1/49
(b) Shuffle a deck, draw a card, and put it back. Shuffle again, draw a card, and put it back. What is the probability that both cards will be a queen?
4/52 x 4/52
(c) An urn has 4 blue marbles and 8 red marbles. Three marbles are drawn randomly without replacement. What is the probability that all three are red?
8/12 x 7/11 x 6/10
(d) Repeat part c, but this time the marbles are drawn with replacement.
8/12 x 8/12 x 8/12
(a) Draw one card from a well-shuffled deck. The chance it will be a spade is 1/4. The chance it will be a club is 1/4. True or false: the chance it will be a club or a spade is 1/4 + 1/4
True, mutually exclusive
(b) Roll a die. What is the chance of rolling 1 or 2?
cannot be both, use addition rule 1/6 + 1/6 = 1/3
A scan for a virus detects the virus with a probability of 0.9. If three independent scans are performed, what is the probability that at least one of them detects the virus
chance at least one detected = 1 - chance of non detected 1 - (0.1 x 0.1 x 0.1)
(a) Roll two dice. Find the probability that at least one will be a 5.
chance neither is 5 = 5/6 x 5/6 = 25/36 chance at least one is 5 = 1 - (25/36) = 11/36
(b) (Related to the paradox of the Chevalier de Méré) In a certain gambling game, a pair of dice is rolled 24 times. You win if you roll "double 1s" at least once. (Double 1s is the event that you roll a 1 on both dice simultaneously.) What is the probability that you win
chance of opposite = (35/36)^24 chance of winning = 1 - (35/36)^24
(f) A deck is well-shuffled. Draw the top two cards. We want the probability that the first card is the ace of spades and the second card is the king of spades. The probability of getting the ace of spades is 1/52, and the probability of getting the king of spades is 1/52. We want the probability of both. True or false: The probability is 1/52 × 1/52
false, dependent, 1/52 x 1/51
(e) Roll two dice. True or false: the chance of getting a 5 on the first die or a 5 on the second die is 1/6 + 1/6
false, too big, not mutually exclusive could get 5 on both dice
A backup hard drive is an example of a parallel component. At least one of the hard drives must work for the data to be safe. Suppose the hard drive's reliability is 0.99 and the backup's reliability is 0.95. What is the reliability of this system?
reliability = 1 - chance both fail = 1 - (0.01 x 0.05)