4.7 A question about dice. Here is a question that a French gambler asked the ma

Question

4.7 A question about dice. Here is a question that a French gambler asked the mathematicians Fermat and Pascal at the very beginning of probability theory: what is the probability of getting at least one 6 in rolling four dice? The Law of Large Numbers applet allows you to roll several dice and watch the outcomes. (Ignore the title of the applet for now.) Because simulation-just like real random phenomena-often takes very many trials to estimate a probability accurately, let's simplify the question: is this probability clearly greater than 0.5, clearly less than 0.5, or quite close to 0.5? Use the applet to roll four dice until you can confidently answer this question. You will have to set "Rolls" to 1 so that you have time to look at the four up-faces. Keep clicking "Roll dice to roll again and again. How many times did you roll four di ce? What percent of your rolls produced at least one 6?

Explanation / Answer

Here, we are rolling 4 dice.

Hence number of Sample points in a Sample space = n(S) = 6^4= 1296.

Now we have to find the probability of getting at least one 6

P( getting at least one 6) = 1- P( not getting 6)

Now number of chances of not getting 6 on 1st die = 5/6

Multiplying by number of chances of not getting 6 on 2nd die 5/6*5/6=25/36

Multiplying by umber of chances of not getting 6 on 3rd die

(5/6) *(5/6) *(5/6) = 125/216

Multiplying by number of chances of not getting 6 on 4th die

(5/6) *(5/6) *5/6) *(5/6) = 625/1296

Thus Probability of not getting 6 in rolling 4 dice= 625/1296= 0.48225

Hence Probability of getting at least one 6 in rolling 4 dice= 1- P( not getting 6 in rolling 4 dice)

= 1- 0.48225

= 0.5177

Which is quite close to 0.5.

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