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Chapter 15: Q9P (page 760)
Use the second method of Problemto show that the expected number of successes inBernoulli trials with probabilityof success is. Hint: What is the expected number of successes in one trial?
Expected number of successes in one trail, i.e., .
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Use Problemto find the expected value of the sum of the numbers on the dice in Problem.
A letter is selected at random from the alphabet. What is the probability that it is one of the letters in the word “probability?” What is the probability that it occurs in the first half of the alphabet? What is the probability that it is a letter after x?
The following problem arises in quantum mechanics (see Chapter , Problem). Find the number of ordered triples of nonnegative integers a, b, c whose sum is a given positive integer n. (For example, if, we could haveoror or or or or .) Hint: Show that this is the same as the number of distinguishable distributions of n identical balls inboxes, and follow the method of the diagram in Example .
Two dice are thrown. Use the sample space (2.4) to answer the following questions.
(a) What is the probability of being able to form a two-digit number greater than
33 with the two numbers on the dice? (Note that the sample point 1, 4 yields
the two-digit number 41 which is greater than 33, etc.)
(b) Repeat part (a) for the probability of being able to form a two-digit number
greater than or equal to 42.
(c) Can you find a two-digit number (or numbers) such that the probability of
being able to form a larger number is the same as the probability of being able
to form a smaller number? [See note part (a)]
Are the following correct non-uniform sample spaces for a throw of two dice? If
so, find the probabilities of the given sample points. If not show what is wrong.
Suggestion: Copy sample space (2.4) and circle on it the regions corresponding to the points of the proposed non-uniform spaces.
(a) First die shows an even number.
First die shows an odd number.
(b) Sum of two numbers on dice is even.
First die is even and second odd.
First die is odd and second even.
(c) First die shows a number≤3.
At least one die shows a number > 3.
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