1 What is the probability that a 7digit phone number contain

1. What is the probability that a 7-digit phone number contains at least one 7? (Repetition of numbers and lead zero are allowed).

2. If the letters in the word POKER are rearranged, what is the probability that the word will begin with the letter O and end with the letter R?

3. Mark draws one card from a standard deck of 52. He receives $ 0.30 for a club, $ 0.60 for an ace and $ 0.95 for the ace of clubs. How much should he pay for one draw?

4. A charity holds a raffle in which each ticket is sold for $25. A total of 8000 tickets are sold. They raffle one grand prize which is a Mercedes Benz E350 valued at $65000 along with 3 second prizes of Honda motorcycles valued at $12000 each. What are the expected winnings for a single ticket buyer? Express to at least three decimal place accuracy in dollar form (as opposed to cents).

5.A raffle has a grand prize of a European cruise valued at $6000 with a second prize of a Las Vegas getaway valued at $700. If each ticket costs $2 and 9400 tickets are sold, what are the expected winnings far a ticket buyer? Express to at least three decimal place accuracy in dollar form (as opposed to cents).

6.A sign on the pumps at a gas station encourages customers to have their oil checked, and claims that one out of 5 cars needs to have oil added. If this is true, what is the probability of each of the following:

A. One out of the next four cars needs oil.

B. Two out of the next eight cars needs oil.

C. 10 out of the next 40 cars needs oil.

Solution

1) P(at least one 7) = 1 - P(no 7\'s)

P(no 7\'s) = 0.9⁹

Therefore, P(at least one 7) = 1 - 0.9⁷ = 0.5217

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2) Total number of ways of arraning letters in word POKER = 5*4*3**2*1 = 120

Now, the word will begin with the letter O and end with the letter R. This can be done using 3*2*1 = 6 ways

Required probability = 6/120 = 0.05

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3) Pay = (12/52)*0.3 + (3/52)*0.6 + (1/53)*0.95 = $ 0.122

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4) expected winning = (1/8000)*65,000 + (3/8000)*12,000 - 25

= -12.375

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5) expected winning = (1/9400)*6000 + (1/9400)*700 - 2

= -1.287

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6)

p = 1/5 = 0.2.

Let X be the cars that need to be oiled.

X follows binaomial distribution

a) P(X = 1) when n= 4 = (4 C 1) * 0.2^1 * 0.8^3 = 0.4096

b) P(X = 2) when n = 8 = (8 C 2) * 0.2^2 * 0.8^6 = 0.2936

c) P(X = 10) when n = 40 = (40 C 10)* 0.2^10* 0.8^30 0.1074