THE TOSS OF A COIN

New Zealand Listener, Volume 3, Issue 59, 9 August 1940, Page 18

THE TOSS OF A COIN

HE main course in this week’s menu for minds is cooked up by that. indefatigable chef from Taupo, H. G. Lambert. In the issue of June 28 he asked what the chances would be of a coin landing heads exactly five times out of ten tosses, and in the issue of July 12 he gave this answer: Exactly 63 times out of 256. He told us he expected some queries, and sure enough they arrived. S. J. S. (Spreydon), voiced our own ideas when he said the answer did not satisfy him, since every toss has a 50/50 chance of being a head. Where, he asked, did the 256 come in? Pennies, we believed, came down whichever way they pleased; but now it seems that their behaviour can be analysed. H.G.L. was asked for an explanation, and obliged, as usual. Here it is: The coin (he says) can land either of two ways in each of ten separate tosses, so that there are 1024 (two to the tenth power) possible ways in which the experiment can result, all equally possible. Now the combination of 0 heads and 10 tails can happen only one way, so that its probability is 1/1024. But the combination of 1 head and 9 tails can happen 10 different ways, so that the probability of it happening at all is 10/1024. The combination of 2 heads and 8 tails can happen, by the same reasoning, 45 different ways, giving a probability of 45/1024. Similarly, the probability of the remaining possible combinations is as follows:

H.G.L. proves his table by noting that the number of ways in which each proportion of heads-to-tails can occur does indeed add up to 1024 (add it and see); and that the probability of the’various proportions coming out adds up to 1 as it should. In the case of 5 heads out of 10 tosses the probability of 252 out of 1024 is by reduction 63 out of 256, which was the answer given. There still remained some points to be cleared up. We could see clearly enough that 10 tails could only come down 1 way, as H.G.L. says; or that 1 head 9 tails, could only come down 10 ways. That is, the head could come into the sequence anywhere in the ten tosses. But it seemed to be more difficult. to work out this variation of sequences with the other combinations. How, for example, did H.G.L. arrive at the statement that the number of ways .2 heads can result in, 10 tosses is 45? Our Worry Anticipated Fortunately, this admirable puzzler anticipated that worry and answered the question in advance. The two heads,

he points out, can occur on the first and second tosses, or the first and third, first and fourth, etc., or on the second and third, second and fourth, etc., or on the third and fourth, third and fifth, etc., etc. In short, the number of possible ways is the number of combinations of 10 things taken two at a time, which, expressed mathematically, is 10! over (10-2)! (2)! when ! is the sign indicating that the figures are factorial. From this H.G.L. elucidates the general formula that the probability of N! out of 2 to the power of N (N-H)! H! equals N! over 2 to the power of N(N-H)! H!. As. the PP found himself able to follow all that (surprisingly enough!),

he gives it to puzzlers, expecting that they will be as interested as he was and as grateful to H.G.L. ior all the trouble. ANSWERS (See issue of July. 26) penser ess Crossword: BY>n PO ney Bane *

The Five Travellers: Since C does not go as far as E, C gets out at Riccarton. Since E goes further than B, B gets out at Papanui. E goes at least as far as Belfast and A as far as Stewart’s Gully. Since E does not get out at Belfast, D must do so. Since E does not go as far —

as A, E gets out at Stewart’s Gully and A at Kaiapoi. — (Problem and answer from R.G.). Clocks: Midnight on Wednesday. — (Problem and answer from R.G.). PROBLEMS Condensed Crossword (Each word is of four letters) CLUES ACROSS: Sobs changed for a top fellow. Palindrome for a girl’s name that is used in India as a means of exchange. When verse does not it is said to have no proper plan. Often seen in cellars, and sometimes the sack is in them in a sack. CLUES DOWN: Put the question and where will you be in the sun? If it happened this it did not happen often. From the fact that she sang she was known as this. Shakespeare was without much French, but he had enough to use this one, Sentry Duty The diagram printed with this is a plan of six posts which have to be guarded by five privates. There is one vacant post. The sergeant of the guard, it seems, has an-i ingenious mind. He rules that the sentries can only move along

the lines indicated, and that only one private can move at a time, At no time must there be more than one private at one post. When A finds himself at the post where B starts, and Bat the post where A starts, the five privates may go off duty. They take their posts as shown at noon. Each man takes 10 minutes to move from one post to another. If no time is wasted, at what time will they go off duty?-(Problem from G. Tisbury).

Along The Waterfront A row of nearly 300 houses was built facing the sea. They were numbered consecutively from end to end. One day, in a contemplative frame of mind, Captain Cook realised that the number of his house was most unusual. If all the numbers before his were added, the sum was identical with that of all the numbers after it. What was the number of his house?-(Problem from Captain Cook). Time For The Guard This is one of several puzzles for

which "Puzzled," Waihi Beach, asks answers: The guard on a train heard the first stroke of 5 from a clock on a bridge which the train was approaching in a straight line. He noted the time as 1 2-5 secs. past 5 by his watch, which was right by the clock. The engine of the train passed under the clock at 13 2-5 secs. past 5, and the guard heard the last stroke of 5 at 30 1-5 secs. past 5. The clock took 29 1-5 secs. to strike 5. If sound travels at the rate of 1100 feet per second, how long was the train?

CORRESPONDENCE 2: Cc. L. Allen (Sumner): Shunts correctly, tings the changes on the bottles of wine, makes Sprinter win by four yards, and sends a problem about the tide which we have used_ before; but thanks all the same. P.J.Q. (Motueka): Weighs the bricks, shunts the trucks, and shifts the bottles, all correctly. Hi. A. Martin (Wairoa): Shows that there is more than one method of shifting the trucks, Puzzled (Waihi Beach): Thinking over those others. L.W.R. (Nelson): Disagrees with R.G.’s answer to his problem of the five travellers, Says that B alights at Riccarton, C at Papanui, D at Belfast, E at Stewart’s Gully; and A at Kaiapoi. Who is correct? G. Tisbury: Thank you for that good problem. As you see, the trucks could be changed over without "slipping."

Heads Tails Chancesin 1024 120 210 252 210 120 45 10 if COMNAUMHW OrPFNWAUNON