positions, the number of arrangements that can be formed is C 83. Thus, P(9) =
C 83 (0.35)
4
(0.65)
5
= 56(0.35)
4
(0.65)
5
= 0.0975.
- Write the equation of Pascal probability distribution
Generalize from step 1 to obtain P(X) = C^^P^l - Pf-*, where P = probability that
event E will occur on a single trial. Here P = 0.35. - Apply the foregoing equation to find the probabilities
corresponding to the given X values
The results are P(4) = 1(0.35)
4
(0.65)° = 0.0150; P(5) = 4(0.35^(0.6S)
1
= 0.0390; P(G) =
10(0.35)
4
(0.65)
2
= 0.0634. Similarly, P(I) = 0.0824; P(S) = 0.0938; P(9) = 0.0975 from
step 1; P(IO) = 0.0951.
Thus, as X increases, P(X) increases until X= 9, and then it decreases. The variable X
can assume an infinite number of values in theory, and the corresponding probabilities
form a converging series having a sum of 1. - Compute the probability that IO or fewer ejections
will be required
Sum the values in step 3; P(X < 10) = 0.4862.
5. Compute the probability that more than 10 ejections
will be required
Since it is certain that X will assume a value of 10 or less or a value of more than 10,
P(X> 1O)-I- 0.4862 = 0.5138. - Compute the average number of ejections required
in the long run
Consider that the process of placing a set of four objects in the receptacle is continued in-
definitely, thereby generating an infinite set of values of X. Since there is a 35 percent
chance that a specific object will enter the receptacle after being ejected from the mecha-
nism, it will require an average of 1/0.35 = 2.86 ejections to place one object in the recep-
tacle and an average of 4(1/0.35) = 11.43 ejections to place four objects in the receptacle.
Thus, the infinite set of values of X has an arithmetic mean of 11.43.
POISSON PROBABILITY DISTRIBUTION
A radioactive substance emits particles at an average rate of 0.08 particles per second. As-
suming that the number of particles emitted during a given time interval has a Poisson
distribution, find the probability that the substance will emit more than three particles in a
20-s interval.
Calculation Procedure:- Compute the average number of particles emitted in 20 s
Let T denote an interval of time, in suitable units. Consider that an event E occurs ran-
domly in time but the average number of occurrences of E in time T 9 as measured over a
relatively long period, remains constant. Let m = average (or expected) number of occur-
rences of E in T 9 and X = true number of occurrences of E in T. The variable X is said to
have a Poisson probability distribution.
In the present case, X= number of particles emitted in 20 S 9 and m = 20(0.08) = 1.6.