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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 audited. Otherwise, she avoids taxes. Suppose a player takes 5 turns at rolling the dice. The probability that she does not get audited is ________.

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MrRoyal

Answer:

[tex]Probability = \frac{243}{1024}[/tex]

Step-by-step explanation:

Given

[tex]Sum = \{7, 11, 12\}[/tex]

[tex]Rolls = 5[/tex]

Required

Probability of not getting audited

If a pair of dice is rolled, the following are the observations of the sum

[tex]Outcomes = 36[/tex]

[tex]Sum\ of\ 7 = 6[/tex]

[tex]Sum\ of\ 11 = 2[/tex]

[tex]Sum\ of\ 12 = 1[/tex]

So, in a single roll; The probability of getting audited is:

[tex]p= \frac{1 + 2 + 6}{36}[/tex]

[tex]p= \frac{9}{36}[/tex]

[tex]p= \frac{1}{4}[/tex]

The probability of not getting audited in a single roll is:

[tex]q = 1 - p[/tex] --- Complement rule

[tex]q = 1 - \frac{1}{4}[/tex]

Take LCM

[tex]q = \frac{4 - 1}{4}[/tex]

[tex]q = \frac{3}{4}[/tex]

The probability of not getting audited in 5 rolls is:

[tex]Probability = q^5[/tex]

[tex]Probability = (\frac{3}{4})^5[/tex]

[tex]Probability = \frac{3^5}{4^5}[/tex]

[tex]Probability = \frac{243}{1024}[/tex]

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