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What is the probability of getting the number 3 at least 5 times if I roll a regular dice 6 times??

What is the probability of getting the number 3 at least 5 times if I roll a regular dice 6 times??

Grade:12

1 Answers

Vasanth SR
askIITians Faculty 1335 Points
6 years ago
Let us first make clear some assumptions.

The random experiment being done is repeatedktimes and each repetition is independent from the others.
On each trial, whether or not we have a "success" is random with probability1n1n.
The quantity of interest (which I'll call the random variableXX) is how many total successes occur.
An experiment obeying these conditions is called abinomial experiment. The random variableXXfollows aBinomial distributionwith parameterskktrials, and1n1nas the probability of success.

So for anym∈{0,1,2,…,k}m∈{0,1,2,…,k}, we can the probability of exactlymmsuccesses as:
P(X=m)=(km)(1n)m(1−1n)k−mP(X=m)=(km)(1n)m(1−1n)k−m

Now, you are interested in at least five successes. Since the eventsX=mX=mandX=bX=bare disjoint for anym≠bm≠b, we can just add the probabilities to get:
P(X≥5)=∑km=5(km)(1n)m(1−1n)k−mP(X≥5)=∑m=5k(km)(1n)m(1−1n)k−m

Ifk≥10k≥10, we can reduce the number of terms we need to compute by using the fact thatP(X≥5)+P(X≤4)=1P(X≥5)+P(X≤4)=1to write:
P(X≥5)=1−∑4m=0(km)(1n)m(1−1n)k−mP(X≥5)=1−∑m=04(km)(1n)m(1−1n)k−m

There isn't much you can do to make this answer simpler, so this is a good stopping point for the exact answer.

Approximation
Now, under some circumstances, we can use theNormal approximation to the binomial distributionto get an answer that is easier to evaluate. In particular, ifkn>>1kn>>1(for example greater than 10) we can observe the following. The expected number of successes isknknand the standard deviation iskn(1−1n)−−−−−−−−√kn(1−1n). So to good approximation,XXfollows a normal distribution with meanknknand standard deviationkn(1−1n)−−−−−−−−√kn(1−1n).

Then the probability thatX≥5X≥5can be well-estimated as:
P(X≥5)≈P⎛⎝⎜Z≥4.5−knkn(1−1n)√⎞⎠⎟P(X≥5)≈P(Z≥4.5−knkn(1−1n))

withZZa random variable with astandard normal distribution.

This probability can be found automatically with many modern calculators. It can be found using a standard normal table. It can be found using online calculators likeNormal Distribution Calculator

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