Example 2. Let us do a similar problem with a die, asking this time for the probability of exactly 3 aces in 5 tosses of the die. If A means ace and N not ace, the probability of a particular sequence such as ANNAA is since the probability of A is , the probability of N is , and the tosses are independent. The number of such sequences containing 3 A's and 2 N's is C(5, 3); thus the probability of exactly 3 aces in 5 tosses of a die is C(5, 3)()³ ()². Generalizing this, we find that the probability of exactly x aces in n tosses of a die is (7.2) f (1) = C(n, a)(¿)*()"-. n-x In the two examples we have just done, we have been concerned with repeated independent trials, each trial having two possible outcomes (h or t, A or N) of given probability. There are many examples of such problems; let's consider a few. A manufactured item is good or defective; given the probability of a defect we want the probability of x defectives out of n items. An archer has probability p of hitting a target; we ask for the probability of x hits out of n tries. Each atom of a radioactive substance has probability p of emitting an alpha particle during the next minute; we are to find the probability that x alpha particles will be emitted in the next minute from the n atoms in the sample. A particle moves back and forth along the x axis in unit jumps; it has, at each step, equal probabilities of Bernoulli Trials
Example 2. Let us do a similar problem with a die, asking this time for the probability of exactly 3 aces in 5 tosses of the die. If A means ace and N not ace, the probability of a particular sequence such as ANNAA is since the probability of A is , the probability of N is , and the tosses are independent. The number of such sequences containing 3 A's and 2 N's is C(5, 3); thus the probability of exactly 3 aces in 5 tosses of a die is C(5, 3)()³ ()². Generalizing this, we find that the probability of exactly x aces in n tosses of a die is (7.2) f (1) = C(n, a)(¿)*()"-. n-x In the two examples we have just done, we have been concerned with repeated independent trials, each trial having two possible outcomes (h or t, A or N) of given probability. There are many examples of such problems; let's consider a few. A manufactured item is good or defective; given the probability of a defect we want the probability of x defectives out of n items. An archer has probability p of hitting a target; we ask for the probability of x hits out of n tries. Each atom of a radioactive substance has probability p of emitting an alpha particle during the next minute; we are to find the probability that x alpha particles will be emitted in the next minute from the n atoms in the sample. A particle moves back and forth along the x axis in unit jumps; it has, at each step, equal probabilities of Bernoulli Trials
A First Course in Probability (10th Edition)
10th Edition
ISBN:9780134753119
Author:Sheldon Ross
Publisher:Sheldon Ross
Chapter1: Combinatorial Analysis
Section: Chapter Questions
Problem 1.1P: a. How many different 7-place license plates are possible if the first 2 places are for letters and...
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Transcribed Image Text:Example 2. Let us do a similar problem with a die, asking this time for the probability of
exactly 3 aces in 5 tosses of the die. If A means ace and N not ace, the probability
of a particular sequence such as ANNAA is since the probability of
A is , the probability of N is , and the tosses are independent. The number of
such sequences containing 3 A's and 2 N's is C(5, 3); thus the probability of exactly
3 aces in 5 tosses of a die is C(5, 3)()³ ()². Generalizing this, we find that the
probability of exactly x aces in n tosses of a die is
(7.2)
f (1) = C(n, a)(¿)*()"-.
n-x
In the two examples we have just done, we have been concerned
with repeated independent trials, each trial having two possible outcomes (h or t,
A or N) of given probability. There are many examples of such problems; let's
consider a few. A manufactured item is good or defective; given the probability
of a defect we want the probability of x defectives out of n items. An archer has
probability p of hitting a target; we ask for the probability of x hits out of n tries.
Each atom of a radioactive substance has probability p of emitting an alpha particle
during the next minute; we are to find the probability that x alpha particles will be
emitted in the next minute from the n atoms in the sample. A particle moves back
and forth along the x axis in unit jumps; it has, at each step, equal probabilities of
Bernoulli Trials
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