Question

- Determine the number of three-digit area codes that can be made from the digits 0-9, assuming the digits can repeat.

- Suppose that there are 15 people in a class. How many ways can the instructor randomly pick three students, if the order doesn’t matter?

-You are playing a game at a local carnival where you must pick a card from a normal 52-card deck. If you pick a face card (jack, queen or king) you get $2. If you pick an ace, you get $5. If you pick any other card, you have to pay $2. What is your expected value for playing this game?

-Suppose that the probability of your favorite baseball player getting a hit at each at-bat is 0.350. Assume that each at-bat is independent of any other at-bat. What is the probability that he bats five times and gets exactly two hits?

Answer 1

Answer to question# 1)

There are total of 10 digits from 0 to 9

Since each number can repeat each position of the code has 10 options

Hence total number of area codes possible = 10*10*10 = 1000

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Answer to question# 2)

If order doesnot natter

When order does not matter, we use the formula of combinations nCr

Number of ways of selecting 3 out of 15 people = 15C3

Number of ways = 15C3 = 15!/(12!*3!)

Number of ways = 15*14*13/(3*2*1) = 455 ways

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Answer to question# 3)

There are 12 face cards , P(face cards) =12/52

There are four ace cards , P(Ace cards) = 4/52

Rest of the cards = 52-12-4 = 36

P(rest cards) = 36/52

.

We got:

Thus expected value = sum of product of x*p = -0.53846

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Answer to question# 4)

P(hit) = p = 0.35

n = 5

we need to find probability for exact 2 hits, hence x = 2

we use formula of binomial probability for it

P(X=x) = nCx * p^x * (1-p)^(n-x)

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On plugging the values we get:

P(x=2) = 5C2 * 0.35^2 * 0.65^3

P(x=2) = 10 * 0.1225 * 0.2746

P(x=2) = 0.336385