The notion of the uniform probability law (or uniform distribution) over the interval
\begin{align} P[B] = \begin{cases} \frac{\text{length of } B}{\text{length of } S}, & \text{if } B \text{ is a subset of } S \
There are many random phenomena for which it appears plausible to assume a uniform probability law. For example, suppose one is tossing a dart at a line marked 0 to 1. If one is always sure to land on the line and if one feels that , then one is led to conclude that the place at which the dart hits the line has a probability function satisfying (5.1) , with
The distribution function
\begin{align} F(x) = \begin{cases} 0, & \text{if } x \leq a \\[2mm] \frac{x-a}{b-a}, & \text{if } a \leq x \leq b \\[2mm] 1, & \text{if } x \geq b. \end{cases} \tag{5.3} \end{align}
By differentiation, the probability density function may be obtained:
\begin{align} f(x) = \begin{cases} \frac{1}{b-a}, & \text{if } a < x < b \\[2mm] 0, & \text{otherwise.} \end{cases} \tag{5.4} \end{align}
From (5.4) it follows that the definition of a uniform probability law given by (5.1) coincides with the definition given by (4.10) . (See (Fig. 5A) )
Example 5A . Waiting time for a train . Between 7 A.M. and 8 A.M. trains leave a certain station at
Solution
We must first find the set
Example 5B . The probability law of the second digit in the decimal expansion of the square root of a randomly chosen number . A number is chosen from the interval 0 to 1 by a random mechanism that obeys a uniform probability law over the interval. What is the probability that the second decimal place of the square root of the number will be the digit 3? Is the digit
Solution
For
or
The length of the interval described by (5.5) is
Hence the probability of the set
In particular,
Exercises
5.1 . The time, measured in minutes, required by a certain man to travel from his home to a train station is a random phenomenon obeying a uniform probability law over the interval 20 to 25. If he leaves his home promptly at 7:05 A.M., what is the probability that he will catch a train that leaves the station promptly at 7:28 A.M.?
Answer
5.2 . A radio station broadcasts the correct time every hour on the hour between the hours of 6 A.M. and 12 midnight. What is the probability that a listener will have to wait less than 10 minutes to hear the correct time if the time at which he tunes in is distributed uniformly over (chosen randomly from) the interval (i) 6 A.M. to 12 midnight, (ii) 8 A.M. to 6 P.M., (iii) 7:30 A.M. to 5:30 P.M., (iv) 7:30 A.M. to 5 P.M?
5.3 . The circumference of a wheel is divided into 37 arcs of equal length, which are numbered 0 to 36 (this is the principle of construction of a roulette wheel). The wheel is twirled. After the wheel comes to rest, the point on the wheel located opposite a certain fixed marker is noted. Assume that the point thus chosen obeys a uniform probability law over the circumference of the wheel. What is the probability that the point thus chosen will lie in an arc (i) with a number 1 to 10, inclusive, (ii) with an odd number, (iii) numbered 0?
Answer
(i)
5.4 . A parachutist lands on the line connecting 2 towns,
5.5 . An angle
Answer
5.6 . A number is chosen from the interval 0 to 1 by a random mechanism that obeys a uniform probability law over the interval. What is the probability that (i) its first decimal will be a 3, (ii) its second decimal will be a 3, (iii) its first 2 decimals will be 3’s, (iv) any specified decimal will be a 3, (v) any 2 specified decimals will be 3’s?
5.7 . A number is chosen from the interval 0 to 1 by a random mechanism that obeys a uniform probability law over the interval. What is the probability that (i) the first decimal of its square root will be a 3, (ii) the negative of its logarithm (to the base
Answer
(i)
