#4 A cash register contains 5 $10 bills and 3 $5 bills. You randomly pick 4 bill
ID: 3271883 • Letter: #
Question
#4A cash register contains 5 $10 bills and 3 $5 bills. You randomly pick 4 bills. a. What is the probability you pick exactly two $10 bills? b. What is the probability you pick at most one $5 bill? The probability of winning the lottery jackpot is 1/M. Players pick six numbers from two separate pools of numbers - five different numbers in any order from 1 to 75, with no replacement, and one number from another pool between 1 and 15. In a different lottery in another state, the probability of winning the jackpot is 1/N where five numbers are drawn out of a drum with 59 numbers (numbered 1 to 59, no replacement) in any order, and one number is drawn from another drum with numbers between I and 35 4.
Explanation / Answer
3 ) a) 5 $10 bills and 3 $5 bills
we pick 4 bills
1 - $10 bills
2 - $5 bills
total = 5 +3 = 8
a) P(X1 = 2) = 5C2 * 3C2 / 8C4 {choose 2 from 5 $10 bills and 2 from 3 $5 bills , nCr is binomial coefficient}
= 10 * 3 / 70
= 3/7
b) P(X2 <= 1) = P(X2 = 0) + P(X2 = 1)
= 5C4 /8C4 + (5C3 * 3C1) /8C4
= (5 + 10 *3)/
= 35/70
= 1/2
4)
"Players may pick six numbers from two separate pools of numbers—five different numbers from 1 to 75 and one number from 1 to 15. You win the jackpot by matching all six winning numbers in a drawing."
In order to win the jackpot, you need to correctly guess five numbers between 1 and 75 from one pool as well as another number between 1 and 15 from a completely separate pool. What's the probability of guessing correctly?
Let's start by thinking about the five numbers drawn from the pool ranging between 1 and 75. How many sequences of five numbers from this pool are possible? Well, there are 75 numbers that can be drawn first. After drawing that initial number, there are 74 numbers that can be drawn second (since the numbers don't go back into the pool after being drawn). There are 73 possible numbers that can be drawn third, 72 that can be drawn fourth, and 71 that can be drawn fifth. This means that there are a total of
75 x 74 x 73 x 72 x 71 = 2,071,126,800
sequences of five balls that can be drawn from the first pool. But the folks at the lottery are generous and actually don't care about the order that you write down your guessed numbers. Since there are 5 x 4 x 3 x 2 x 1 = 120 different possible arrangements of the order in which those five balls were drawn (and since any of those sequences are counted as a match), there are actually a total of
2,071,126,800 / 120 = 17,259,390
possible order-independent combinations of five balls drawn from this pool.
But there's still that sixth ball from the second pool to worry about. Since there are 15 possible numbers in that pool, the total number of possible combinations of all six balls in the Mega Millions lottery game is:
17,259,390 x 15 = 258,890,850
And that is a really big number. Which is bad news for hopeful players because the chances of winning are 1 over this number. In other words, the probability of winning the Mega Millions jackpot is 1 in 258,890,850 or about 0.0000004%—not so good.
In short M= 75*74*73*72*71/120*15 = 258890850
similarly N = 59*58*57*56*55/120*35 = 175223510
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