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) = ^{n}C_{r}p^{r}q^{n-r}

Here, n = 6,

&

…(i)

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.

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