Two Dice Makes It Harder?

Probability Level 2

Anna and Jason play a game of dice, where two fair 6-sided dice are rolled. If the largest face value is either 1, 2, 3 or 4, Anna wins. If the largest face value is 5 or 6, Jason wins. What is the probability that Anna will win?

\frac{1}{3}

\frac{4}{9}

\frac{1}{2}

\frac{5}{9}

\frac{2}{3}

2 solutions

Armain Labeeb
Jun 9, 2016

Relevant wiki: Probability - By Outcomes

The answer is $\frac{4}{9}$ . As the dice are fair , there will be $36$ outcomes in $36$ rolls. Consider the table below: As you can see Anna wins $\frac{16}{36}$ of the possible outcomes and Jason wins $\frac{20}{36}$ of them. $\frac{16}{36}$ can be simplified to $\frac{4}{9}$ , which is the correct answer.

Another explanation is that the probability of the first die rolling a 1,2,3, or 4 is 2/3 and the probability of the second die rolling a 1,2,3 or 4 is also 2/3. Since the two events are independent of each other, the probability of both occurring, and thus Anna winning, is the two probabilities multiplied together. $\frac{2}{3} * \frac {2}{3} = \frac{4}{9}$

Jake Flom - 5 years ago

If the highest number among the two numbers is either 1,2,3 or 4, Anna wins

Would 1,1 be winning for Anna? Both these numbers are equal!

I had left these and similar outcomes out of the solution space, hence 1/3 was my answer!

Salim Pa - 5 years ago

Therefore, both of them are the highest! And therefore, 1,1 would be winning for Anna. Just like frequency tables, there can be two or more highest frequencies.

Armain Labeeb - 5 years ago

Why does it say there will be no same roll in 36 rolls? If they throw the dice 36 times, both will have probably won atleast once.

sven Uden - 5 years ago

@sven Uden I meant to say that the dice are fair and there will be no same outcome in the all 36 possible rolls. For shot, I meant to say the dice are fair.

Armain Labeeb - 5 years ago

Kmd
Jun 11, 2016

Question: why is it necessary to state: "Presuming that the dice does not show the same result twice in 36 rolls"?

Otherwise, solution seems straightforward 4^2/36 = 4/9.

Thanks. I have edited the problem accordingly.

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Calvin Lin Staff - 5 years ago

Two Dice Makes It Harder?

2 solutions

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