Question

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1. Suppose that you and a friend are playing cards and you decide to make a friendly wager. The bet is that you will draw two cards with replacement from a standard deck of 52 cards. If both cards are hearts, you win $25. Otherwise, you have to pay your friend $2. (a) Find the probability distribution for the winning of your bet, say X. Hint: let x be your winning and it can take on possible values 25 and -2. (b) What is the expected value of the bet? (c) If this same bet is made 1000 times, how much would you expect to win or lose? (d) Compute the variance for X. (e) Compute the standard deviation for X. Note: The standard deviation is a measure for risk of the game.

2. The probability of a plant surviving in Kerry’s garden is 0.7. If she plants 19 new plants this year, (a) what is the probability that at least 10 of them survive? (b) What is the probability that exactly 15 survive? (c) What is the probability that less than 9 survive? (d) What is the probability that no more than 12 survive? (e) What is the probability that no less than 10 and no more than 15 survive?

Answer 1

Solution : ( 1 )

Outcome probability payoff

Both cards hearts = (13/52)(12/51) = 0.05882 + 25

Other = 1 - 0.05882 = 0.94118 - 2

( b )

Expected value is 0.05882(25) + 0.94118(-2) = - $0.41186

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( c )

Making this bet 1000 times, you would expect to lose 1000(- 0.41186) = - $411.86

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Solution : ( 2 ).

( a )

Use a binomial table with n = 19 , p = 0.7 and look at the probability table

Then likelihood of n <10 is 0.0324

Hence, the probability that at least 10 of them survive = 1 - 0.0324 = 0.9676

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