Page 353 - DECO504_STATISTICAL_METHODS_IN_ECONOMICS_ENGLISH

Statistical Methods in Economics

Notes a student is enrolled for at least one of the courses, the (conditional) probability that
he is enrolled for French is
30
40 = 0.75
Discrete Sample Space

So far we considered cases where the sample space contains a finite number of points. We consider
the following example where this is not the case.
Example 15: A coin is tossed until a head appears. Describe the sample space. Find the probability
that the coin will be tossed (a) exactly 4 times (b) at the most, 4 times. (c) What is the
probability that head will appear if the coin is tossed an infinite number of times ?
The head may appear at the
(i) very first throw (H)
(ii) second throw, the first toss resulting in a tail (TH)
(iii) third throw, the first two tosses resulting in tails (TTH)
(iv) fourth throw, the first three tosses resulting in tails (TTTH)
and so on: an infinite number of throws may be needed to get a head. The sample
space consists of an infinite number of the sample points
H, TH, TTH, TTTH, TTTTH, ....
The trials are independent. Assume that the coin is fair (unbiased).

1
The probability of the event H =
2

1
11 ⎛⎞ 2
The probability of the event TH = ⋅ = ⎜⎟
22 ⎝⎠
2
1
1
⎛⎞ 2 1 ⎛⎞ 3
The probability of the event TTH = ⎜⎟ ⋅ = ⎜⎟
2
⎝⎠ 2 ⎝⎠
2
1
1
⎛⎞ 3 1 ⎛⎞ 4
The probability of the event TTTH = ⎜⎟ ⋅ = ⎜⎟
⎝⎠ 2 ⎝⎠
2
2
and so on.
1
⎛⎞ 4
(a) The probability that to get a head, the coin will be tossed exactly 4 times is ⎜⎟ .
⎝⎠
2
(b) The event that the coin will be tossed at most 4 times is a compositive event
comprising of the 4 sample events H, TH, TTH and TTTH. Thus the required
probability
1
1
1 ⎛ 1 ⎞ 2 ⎛⎞ 3 ⎛⎞ 4 15
= + ⎜ ⎟ + ⎜⎟ + ⎜⎟ = .
2
2 ⎝ 2 ⎠ ⎝⎠ ⎝⎠ 16
2
(c) Suppose that the coin is tossed as many times as is necessary to get a head. The
required probability is
1
1 + ⎛⎞ 2 + ⎛⎞ 3 +
1
2
2 ⎜⎟ ⎜⎟ ...
⎝⎠
⎝⎠
2
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