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Fig. 6-1. Observed sequences of heads and tails in three games of
30 tosses each.
* After the first three games, the experiment was actually done by shaking 30 pennies
violently in a box and then counting the number of heads that showed.
--- Trang 123 ---
Table 6-1
Number of heads in successive trials of 30 tosses of a coïỉn.
11 16 17 15 17 16 19 18 15 13
11 17 17 12 20 23 11 16 17 14
16 12 15 10 18 17 13 15 14 lỗ
16 12 11 22 12 20 12 lỗ 16 12
16 10 15 13 14 16 15 16 13 18 100 trial
14 14 13 16 15 19 21 14 12 lỗ may
16 11 16 14 17 14 11 16 17 16
19 15 14 12 18 l5 14 21 11 16
17 17 12 13 14 17 9 13 19 13
14 12 15 17 14 10 17 17 12 11
H \ |>——— OBSERVED IN THIS
; \ EXPERIMENT
NUMBER OF / `
GAMES IN ; :
WHICH THE_ 10 / \
SCORE WAS H \
OBTAINED r \
j \„—PROBABLE NUMBER
5 , Ñ
0 -. TẾ
0 5 10 † 20 25 30
k = NUMBER OF HEADS
Fig. 6-2. Summary of the results of 100 games of 30 tosses each.
The vertical bars show the number of games In which a score of k heads
was obtained. The dashed curve shows the expected numbers of games
with the score k obtained by a probability computation.
--- Trang 124 ---
Looking at the numbers in Table 6-1, we see that most of the results are
“near” 15, in that they are between 12 and 18. We can get a better feeling for
the details of these results if we plot a graph of the đ¿str?bufion of the results.
W© count the number of games in which a score of k was obtained, and plot this
number for each &. Such a graph is shown in Fig. 6-2. A score of 15 heads was
obtained in 13 games. A score of 14 heads was also obtained 13 times. Scores of
16 and 17 were each obtained more than 13 times. Are we to conclude that there
is some bias toward heads? Was our “best estimate” not good enough? Should
we conclude now that the “most likely” score for a run of 30 tosses is really
16 heads? But waitl In all the games taken together, there were 3000 tosses.
And the total number of heads obtained was 1493. The fraction of tosses that
gave heads is 0.498, very nearly, but slightly iess than half. We should certainly
no‡ assume that the probability of throwing heads is greater than 0.5! "The fact
that one øarticular set of observations gave 16 heads most often, is a fÏuctuation.
We still expect that the rmost likely number of heads is 1ã.
W©e may ask the question: “What ¿s the probability that a game of 30 tosses
will yield 15 heads——or 16, or any other number?” We have said that in a game
of one toss, the probability of obtaining øne head is 0.5, and the probability of
obtaining no head is 0.5. In a game of two tosses there are ƒour possible outcomes:
HH, HT, TH, TT. Since each of these sequences is equally likely, we conelude
that (a) the probability of a score of two heads is +, (b) the probability of a score
of one head is 2, (c) the probability of a zero score is +. There are /o ways oŸ
obtaining one head, but only one of obtaining either zero or bwo heads.
Consider now a game of 3 tosses. The third toss is equally likely to be heads
or tails. There is only one way to obtain 3 heads: we znusứ have obtained 2 heads
on the first two tosses, and then heads on the last. "There are, however, £hree
ways of obtaining 2 heads. We could throw tails after having thrown two heads
(one way) or we could throw heads after throwing only one head in the frst two
%osses (two ways). So Íor scores of 3-J, 2-H, 1-H, 0-H we have that the number
of equally likely ways 1s 1, 3, 3, 1, with a total of 8 diferent possible sequences.
'The probabilities are 8) ẩ› Š› §-
The argument we have been making can be summarized by a diagram like
that in Fig. 6-3. It is clear how the diagram should be continued for games with
a larger number of tosses. Figure 6-4 shows such a diagram for a game of 6 tosses.
The number of “ways” to any point on the diagram is just the number of diferent
“paths” (sequences of heads and tails) which can be taken from the starting point.
The vertical position gives us the total number of heads thrown. “The set of
--- Trang 125 ---
WAYS WAYS WAYS SCORE_ PROB.
H 1 3H 1/8
n1 h 3 2H 3/8
< SỰ 2S
.. „>7 1H 3/8
FIRST ị ' 1 0H 1/8
TOSS SECOND Ị
'TOSS 'THIRD
Fig. 6-3. A diagram for showing the number of ways a score of 0, 1,
2, or 3 heads can be obtained In a game of 3 tosses.
SCORE
1 4 15 4
1 3 10
< 2 6 20 3
1 3 10
1 4 15 2
Fig. 6-4. A diagram like that of Fig. 6-3, for a game of 6 tosses.
numbers which appears in such a diagram is known as Pascals triangle. The
numbers are also known as the Ùznormal coefficien‡s, because they also appear In
the expansion of (ø + Ö)”. If we call ø the number of tosses and k the number of
heads thrown, then the numbers in the diagram are usually designated by the
symbol (): W©e may remark in passing that the binomial coeflicients can also be
computed from
LÀN nÌ (6.4)
kj — kl{n— k)!' l
where ml, called “n-factorial,” represents the produet (n)(w®— 1)(m—2) - - - (3)(2)(1).
W© are now ready to compute the probability P{k,m) of throwing k heads in