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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) = nCrprqn-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, the probability of getting 3 successes is 1/2.