3Brief on Bell Shapes

Before expanding the previous Statistical Scenario let's briefly explore why the histogram, reproduced in Figure 3.1, is shaped the way it is: bell-shaped. It tapers off symmetrically on each side from a single peak in the middle.

A bar graphical representation 1000 people each flip a fair coin 10 times, where frequency out of 1000 is plotted on the y-axis on a scale of 0–300 and number of heads on the x-axis on a scale of 0–10.

Figure 3.1

Since each coin flip has two possible outcomes and we are considering ten separate outcomes together, there are a total of img unique possible patterns (permutations) of heads and tails with 10 flips of a coin. Of these, there is only one with 0 heads and only one with 10 heads. These are the least likely outcomes.

TTTTTTTTTT HHHHHHHHHH

There are ten with 1 head, and ten with 9 heads:

HTTTTTTTTT THHHHHHHHH
THTTTTTTTT HTHHHHHHHH
TTHTTTTTTT HHTHHHHHHH
TTTHTTTTTT HHHTHHHHHH
TTTTHTTTTT HHHHTHHHHH
TTTTTHTTTT HHHHHTHHHH
TTTTTTHTTT HHHHHHTHHH
TTTTTTTHTT HHHHHHHTHH
TTTTTTTTHT HHHHHHHHTH
TTTTTTTTTH HHHHHHHHHT

Since there are 10 times more ways to get 1 or 9 heads than 0 or 10 heads, we expect to flip 1 or 9 heads 10 times more often than 0 or 10 heads.

Further, there are 45 ways to get 2 or 8 heads, 120 ways to get 3 or 7 heads, and 210 ways to get 4 or 6 heads. Finally, there are 252 ways to get 5 heads, which ...

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