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alisha [4.7K]
3 years ago
10

In a game called taxation and evasion, a player rolls a pair of dice. if on any turn the sum is 7, 11, or 12, the player gets au

dited. otherwise, she avoids taxes. suppose a player takes 5 turns at rolling the dice. the probability that she gets audited no more than 2 times is
Mathematics
1 answer:
bezimeni [28]3 years ago
5 0

The probability that she gets audited no more than 2 times is 0.896482...

As, a pair of dice are rolling here, so the number of total possible outcome = (6×6) = 36

For Sum = 7 , the favorable outcomes are: (1,6) (2,5) (3,4) (4,3) (5,2) and (6,1)

For Sum = 11 , the favorable outcomes are: (5,6) and (6,5)

For Sum = 12 , the favorable outcome is: (6,6)

Probability = (Number of favorable outcomes)÷(Number of total outcomes)

So, P(7)= \frac{6}{36} =\frac{1}{6}

P(11) = \frac{2}{36} = \frac{1}{18}

P(12) = \frac{1}{36}

P( 7 or 11 or 12) = \frac{1}{6}+\frac{1}{18}+\frac{1}{36} = \frac{6+2+1}{36}=\frac{9}{36} = \frac{1}{4}

Here the total number of trials = 5 and the probability of getting audited = \frac{1}{4}

According the binomial distribution formula:

P(X) = (ⁿCₓ )(P)ˣ (1-P)ⁿ⁻ˣ

where P(X) is the probability of x successes out of n trials

Here, n= 5 and P = 1/4 and we need to find the probability of getting audited no more than 2 times. This means she can gets audited 0, 1 or 2 times.

So,

P(X=0)+P(X=1) +P(X=2)\\\\= [^5C^0 (\frac{1}{4})^0 (1-\frac{1}{4})^5^-^0 ]+[^5C^1 (\frac{1}{4})^1 (\frac{3}{4})^5^-^1]+[^5C^2 (\frac{1}{4})^2 (\frac{3}{4})^5^-^2]\\\\=  (\frac{3}{4})^5+ (5) (\frac{1}{4}) (\frac{3}{4})^4  +(10)(\frac{1}{4})^2 (\frac{3}{4})^3\\\\=0.237304... +0.395507...+0.263671...\\\\ = 0.896482...

So, the probability that she gets audited no more than 2 times is 0.896482...

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Answers:

  • <u>24000 dollars</u> invested at 4%
  • <u>18000 dollars</u> was invested at 7%

======================================================

Work Shown:

x = amount invested at 4%

If she invests x dollars at 4%, then the rest (42000-x) must be invested at the other rate of 7%

She earns 0.04x dollars from that first account and 0.07(42000-x) dollars from the second account

This means we have

0.04x+0.07(42000-x)

0.04x+0.07*42000-0.07x

0.04x+2940-0.07x

-0.03x+2940

This represents the total amount of money earned after 1 year.

We're told the amount earned in interest is $2220, so we can say,

-0.03x+2940 = 2220

-0.03x = 2220-2940

-0.03x = -720

x = -720/(-0.03)

x = 24000 dollars is the amount invested at 4%

42000-x = 42000-24000 = 18000 dollars was invested at 7%

----------------------

As a check, we can see that

18000+24000 = 42000

and also

0.04x = 0.04*24000 = 960 earned from the first account

0.07*18000 = 1260 earned from the second account

1260+960 = 2220 is the total interest earned from both accounts combined

This confirms our answers.

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