Lenny D. answered • 01/20/20

Former professor at Tufts University with decades on Wall Street

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 2^{3} 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 p^{2} and the probability of it happening three times in a row is p^{3}

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.