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Chapter 15: Q13MP (page 777)
A radioactive source emits particles during an observation lasting 10 hours. In how many one-minute intervals do you expectno? ?
The values required is given below.
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(a) A loaded die has probabilitiesof showing 1, 2, 3, 4, 5, 6.What is the probability of throwing two 3’s in succession?
(b) What is the probability of throwing a 4 the first time and not a 4 the second
Time with a die loaded as in (a)?
(c) If two dice loaded as in (a) are thrown, and we know that the sum of the
numbers on the faces is greater than or equal to 10, what is the probability
That both are 5s?
(d) How many times must we throw a die loaded as in (a) to have probability greater than of getting an ace?
(e) Adie, loaded as in (a), is thrown twice. What is the probability that thenumber on the die is even the first timethe second time?
(a) Three typed letters and their envelopes are piled on a desk. If someone puts theletters into the envelopes at random (one letter in each), what is theprobabilitythat each letter gets into its own envelope? Call the envelopes A, B, C, and thecorresponding letters a, b, c, and set up the sample space. Note that “a in A,b in B, c in A” is one point in the sample space.
(b) What is the probability that at least one letter gets into its own envelope?
Hint: What is the probability that no letter gets into its own envelope?
(c) Let A mean that a got into envelope A, and so on. Find the probability P(A)that a got into A. Find P(B) and P(C). Find the probability P(A + B)that either a or b or both got into their correct envelopes, and the probabilityP(AB) that both got into their correct envelopes. Verify equation (3.6).
Set up an appropriate sample space for each of Problems 1.1 to 1.10 and use itto solve the problem. Use either a uniform or non-uniform sample space or try both.
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?
A trick deck of cards is printed with the hearts and diamonds black, and the spades and clubs red. A card is chosen at random from this deck (after it is shuffled). Find the probability that it is either a red card or the queen of hearts. That it is either a red face card or a club. That it is either a red ace or a diamond.
Given a non uniform sample space and the probabilities associated with the points, we defined the probability of an event A as the sum of the probabilities associated with the sample points favorable to A. [You used this definition in Problem 15with the sample space (2.5).] Show that this definition is consistent with the definition by equally likely cases if there is also a uniform sample space for the problem (as there was in Problem 15). Hint: Let the uniform sample space have n<Npoints each with the probability N-1. Let the nonuniform sample space have n points, the first point corresponding to N1 points of the uniform space, the second to N2 points, etc. What is N1 + N2 + .... Nn ?What are p1, p2, ...the probabilities associated with the first, second, etc., points of the nonuniform space? What is p1 + p2 +....+ pn? Now consider an event for which several points, say i, j, k, of the nonuniform sample space are favorable. Then using the nonuniform sample space, we have, by definition of the probability p of the event, p = pi + pj + pk . Write this in terms of the N’s and show that the result is the same as that obtained by equally likely cases using the uniform space. Refer to Problem 15as a specific example if you need to.
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