Five dice are thrown simultaneously. If the occurrence of 3, 4 or 5 in a single die is considered a success, find the probability of at least 3 successes.

Let p denote the probability of getting 3, 4 or 5 in a throw of dice.

Let us find out the value of p.

We know, a dice has 6 faces numbered 1, 2, 3, 4, 5 and 6.

So, the probability of getting a 3, 4 or 5 is given as,

If p denotes the probability of getting success, then let q denote the probability of not getting success.

We can say,

p + q = 1

⇒ q = 1 – p

Let X denote the number of successes in the throw of five dice simultaneously.

Let there be total n number of throws of five dice simultaneously.

Then, the probability of getting r successes out of n throws of dice is given by,

P (X = r) = ^{n}C_{r}p^{r}q^{n-r}

Now, substitute the value of p and q in the above equation.

Also, put n = 5 (Since there are 5 dice throw)

Now, the probability of getting at least 3 successes is given by

Probability of getting at least 3 successes = P(X = 3) + P(X = 4) + P(X = 5)

Thus,

Thus, the probability of getting 3 successes is 1/2.

27