Assuming that our random variable Y belongs to Binomial distribution and Y∼ Binomial (n = 36, p = 1/3). Find the probability that Y > 14. 

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In a Binomial Distribution (discrete) mean=np and variance=np(1-p), so standard deviation=√(np(1-p)).

In this case mean, μ=⅓(36)=12 and standard deviation, σ=√(12×⅔)=√8=2√2=2.828 approx.

The Binomial Distribution is approximately normal (continuous distribution), so Z=(14-12)/(2√2)=√2/2=0.707 approx. N(0.707)=0.76 approx, this is N(p<76%), therefore N(p>76%)=100-76=24%. (Y>14 means that in a trial of 36 attempts there would be specifically 15, 16, 17, ..., 35, or 36 successes. On average we would expect 12 successes out of 36 attempts. A normal distribution is not discrete, so there is a continuous range of values between 14 and 36, not just integers between 15 and 36.)

Using a Binomial Distribution calculator for cumulative probability B(n,p) for Y≤14=0.8129 (81.29%), so the probability for Y>14 is 1-0.8129=18.71%.

(An example of an experiment to which a Binomial Distribution can be applied would be spinning a triangular spinner 36 times. The spinner's sides are coloured red, white and blue. Success occurs when the spinner lands on, say, the white side. So Y>14 would mean more than 14 "white" spins when the trial is over.)

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