percentage of population and probability of coin toss

Question

hello there,

I would to help me how to solve these two problems.

1- Assume the world human population heights follow a perfect normal distribution with standard deviation of 20 and mean of 150, what percentage of the population would have a height between 130 and 170?

2- Having a fair coin i.e. P(Head) = P(Tail) = 0.5, what is the probability of getting a sequence of 3 consecutive tails?

thank you in advance

Answer 1

For the First One, 170 is 1 standard deviation above the mean and 130 is 1 standard devation blow the mean. If the go to a standard normal table, The Probability of being below 170 or 1 standard deviation above the mean is = 8413 and the Probability of being 1 standard deviation below the mean (130 ) is ..1586. The probability of being in between is simply the difference. or .682. Simply put, Is a random variable is normally distributed, Roughly 68% time you will find find it between +-1 standard deviation of the mean. Roughly 95% the time .9544 you will find it between 2 Standard deviation. These relationships are good to memorize

For the second question, there are several ways to answer this and they are all equivalent. The first, if we toss a coin 3 times there are 2³ or 8 possible outcomes. only 1 has all three being tails so 1/8 is the probability (we are implicitly assuming a fair coin when we do this..

More Generally, If the Probability of an event happening is p, The probability of happening twice in a row is p² and the probability of it happening three times in a row is p³

In our example p = 1/2 so the probability of 3 tails is (1/2)³= 1/8. If it was an unfair coin Heads came up 3/4 of the time then the probability of getting tails once is 1/4 the the probability of getting tails 3 times in a row becomes much more remote (1/4)³ = =1/64.