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An experiment succeeds twice as often as it fails. Find the probability that in the next 6 trials there will be at least 4 successes.
Let p be the probability of success in the experiment. Then, q is the probability of failure in the experiment.
According to the question,
An experiment succeeds twice as often as it fails.
⇒ p = 2q
But p + q = 1
⇒ 2q + q = 1 [∵ p = 2q]
⇒ 3q = 1
Then, p = 2q
Let X be a random variable representing the number of successes out of 6 experiments.
Then, the probability of getting r success out of n experiments is given by,
P (X = r) = nCrprqn-r
Here, n = 6,
Then, probability that there will be at least 4 successes out of 6 is given by,
Probability = P (X = 4) + P (X = 5) + P (X = 6)
Put r = 4, 5 and 6 respectively for P (X = 4), P (X = 5) and P (X = 6) in equation (i), we have
⇒ Probability = 0.68
∴, the required probability that in the next 6 trials there will be at least 4 successes is 0.68.