Question

1. Let S = {1, {}, c} be a sample space. List all possible events. 2.Let ? be a sample space and ? be a probability. Prove that there can’t exist events E, F that satisfy ?(E F) = 1/3, ?(E ? F) = 1/2, and ?((E ? F)?) = 3/4. 3. Let A, B, C be events in a sample space S. Prove that (a) ?(A ? B) ? ?(A) + ?(B), (b) ?(A ? B ? C) = ?(A) + ?(B) + ?(C) ? ?(A ? B) ? ?(A ? C) ? ?(B ? C) + ?(A ? B ? C). 4. Your baking cupboard contains 1 cup of whole wheat flour, 1 cup of white flour, 1 cup of brown sugar, 1 cup of white sugar, and 2 eggs. Consider the following random baking experiment: You thoroughly mix three randomly chosen ingredients in a bowl and throw it into the oven. (a) Write down the sample space of this experiment. (b) What is the probability that you will actually bake a cake? 5. Assuming a fair poker deal, what is the probability of a (a) royal flush (b) straight flush (c) flush (d) straight (e) two pair See https://en.wikipedia.org/wiki/List_of_poker_hands for the definition of these poker hands. 6. How many ways are there to deal 52 standard playing cards to four players (so that each player gets 13 cards)? Suppose you are world champion in card dealing, and can deal 52 cards in just one second. Compare the time it would need you to deal all possible combinations with the current age of the universe. 7. We toss a fair die four times. What is the probability that all tosses produce different outcomes? 8*. Prove that the number of unordered sequences of length k with elements from a set X of size n is (n+k-1 over k). Hint: For illustration, first consider the example n = 4, k = 6. Let the 4 elements of the set X be denoted a, b, c, d. Argue that any unordered sequence of size 6 consisting of elements a, b, c, d can be represented uniquely by a symbol similar to ??|·|··|?, corresponding to the sequence aabccd. Now count the number of choices for the vertical bars.

          1. Let S = {1, {}, c} be a sample space. List all possible events.
2.Let ? be a sample space and ? be a probability. Prove that there can’t exist events E, F that satisfy
?(E  F) = 1/3, ?(E ? F) = 1/2, and ?((E ? F)?) = 3/4.
3. Let A, B, C be events in a sample space S. Prove that
(a) ?(A ? B) ? ?(A) + ?(B),
(b) ?(A ? B ? C) = ?(A) + ?(B) + ?(C) ? ?(A ? B) ? ?(A ? C) ? ?(B ? C) + ?(A ? B ? C).
4. Your baking cupboard contains 1 cup of whole wheat flour, 1 cup of white flour, 1 cup of brown sugar, 1 cup of white sugar, and 2 eggs. Consider the following random baking experiment: You thoroughly mix three randomly chosen ingredients in a bowl and throw it into the oven.
(a) Write down the sample space of this experiment.
(b) What is the probability that you will actually bake a cake?
5. Assuming a fair poker deal, what is the probability of a
(a) royal flush
(b) straight flush
(c) flush
(d) straight
(e) two pair
See https://en.wikipedia.org/wiki/List_of_poker_hands for the definition of these poker hands.
6. How many ways are there to deal 52 standard playing cards to four players (so that each player gets 13 cards)? Suppose you are world champion in card dealing, and can deal 52 cards in just one second. Compare the time it would need you to deal all possible combinations with the current age of the universe.
7. We toss a fair die four times. What is the probability that all tosses produce different outcomes?
8*. Prove that the number of unordered sequences of length k with elements from a set X of size n is (n+k-1 over k).
Hint: For illustration, first consider the example n = 4, k = 6. Let the 4 elements of the set X be denoted a, b, c, d. Argue that any unordered sequence of size 6 consisting of elements a, b, c, d can be represented uniquely by a symbol similar to ??|·|··|?, corresponding to the sequence aabccd. Now count the number of choices for the vertical bars.

1. Let S = 1, , c be a sample space. List all possible events.
2.Let ? be a sample space and ? be a probability. Prove that there can’t exist events E, F that satisfy
?(E F) = 1/3, ?(E ? F) = 1/2, and ?((E ? F)?) = 3/4.
3. Let A, B, C be events in a sample space S. Prove that
(a) ?(A ? B) ? ?(A) + ?(B),
(b) ?(A ? B ? C) = ?(A) + ?(B) + ?(C) ? ?(A ? B) ? ?(A ? C) ? ?(B ? C) + ?(A ? B ? C).
4. Your baking cupboard contains 1 cup of whole wheat flour, 1 cup of white flour, 1 cup of brown sugar, 1 cup of white sugar, and 2 eggs. Consider the following random baking experiment: You thoroughly mix three randomly chosen ingredients in a bowl and throw it into the oven.
(a) Write down the sample space of this experiment.
(b) What is the probability that you will actually bake a cake?
5. Assuming a fair poker deal, what is the probability of a
(a) royal flush
(b) straight flush
(c) flush
(d) straight
(e) two pair
See https://en.wikipedia.org/wiki/Listofpokerhands for the definition of these poker hands.
6. How many ways are there to deal 52 standard playing cards to four players (so that each player gets 13 cards)? Suppose you are world champion in card dealing, and can deal 52 cards in just one second. Compare the time it would need you to deal all possible combinations with the current age of the universe.
7. We toss a fair die four times. What is the probability that all tosses produce different outcomes?
8*. Prove that the number of unordered sequences of length k with elements from a set X of size n is (n+k-1 over k).
Hint: For illustration, first consider the example n = 4, k = 6. Let the 4 elements of the set X be denoted a, b, c, d. Argue that any unordered sequence of size 6 consisting of elements a, b, c, d can be represented uniquely by a symbol similar to ??|·|··|?, corresponding to the sequence aabccd. Now count the number of choices for the vertical bars.

Added by Jacqueline G.

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