. Suppose in an event of fire, the probability is 95% that fire alarm will ring in a building. Let x denote the number of fire alarm that will ring in case of fire. Suppose there are 3 fire alarms.
a) Write the probability distribution of x.
b) Explain x=0, x=1, x=2 and x=3
e) What is the probability that
i) exactly 2 alarm will ring ii) at least 2 alarm will ring iii) at most 2 alarm will ring iv) all 3 alarm will ring.
a) The probability distribution of x can be represented as follows:
x = 0: P(x=0) = (1 - P(fire alarm rings))^3 = (1 - 0.95)^3 = 0.0025
x = 1: P(x=1) = 3 * P(fire alarm rings) * (1 - P(fire alarm rings))^2 = 3 * 0.95 * (1 - 0.95)^2 = 0.1425
x = 2: P(x=2) = 3 * (P(fire alarm rings))^2 * (1 - P(fire alarm rings)) = 3 * 0.95^2 * (1 - 0.95) = 0.4275
x = 3: P(x=3) = (P(fire alarm rings))^3 = 0.95^3 = 0.857375
b)
x = 0: Represents the scenario where none of the fire alarms ring in case of fire.
x = 1: Represents the scenario where exactly one fire alarm rings in case of fire.
x = 2: Represents the scenario where exactly two fire alarms ring in case of fire.
x = 3: Represents the scenario where all three fire alarms ring in case of fire.
e)
i) The probability that exactly 2 alarms will ring (P(x=2)) is 0.4275.
ii) The probability that at least 2 alarms will ring is equal to the sum of probabilities of x=2 and x=3. P(at least 2 alarms) = P(x=2) + P(x=3) = 0.4275 + 0.857375 = 1.284875.
iii) The probability that at most 2 alarms will ring is equal to the sum of probabilities of x=0, x=1, and x=2. P(at most 2 alarms) = P(x=0) + P(x=1) + P(x=2) = 0.0025 + 0.1425 + 0.4275 = 0.5725.
iv) The probability that all 3 alarms will ring (P(x=3)) is 0.857375.
a) To write the probability distribution of x, we need to determine the probability of each possible value of x.
Given that there are 3 fire alarms, the possible values for x are 0, 1, 2, and 3.
b) To explain each value of x:
- x = 0: This means that none of the fire alarms ring in case of fire.
- x = 1: This means that only one fire alarm rings in case of fire.
- x = 2: This means that two fire alarms ring in case of fire.
- x = 3: This means that all three fire alarms ring in case of fire.
e) To calculate the probability of each scenario, we need to use the given probability that a fire alarm will ring in a building, which is 95% or 0.95.
i) Probability of exactly 2 alarms ringing:
To calculate the probability that exactly 2 alarms will ring, we use the binomial probability formula:
P(x=k) = (n choose k) * p^k * (1-p)^(n-k)
n = total number of trials = 3 (since there are 3 alarms)
k = number of successful trials (2 alarms ringing)
p = probability of a successful trial (0.95, as given)
(1-p) = probability of an unsuccessful trial (1-0.95 = 0.05)
Plugging the values into the formula:
P(x=2) = (3 choose 2) * 0.95^2 * 0.05^(3-2)
Using the combination formula, (3 choose 2) = 3:
P(x=2) = 3 * 0.95^2 * 0.05
Simplifying the equation, we find the probability that exactly 2 alarms will ring.
ii) Probability of at least 2 alarms ringing:
To calculate the probability that at least 2 alarms will ring, we need to find the probability of exactly 2 alarms ringing, plus the probability of all 3 alarms ringing.
P(at least 2 alarms) = P(x=2) + P(x=3)
We have already calculated P(x=2), so we just need to find P(x=3):
P(x=3) = (3 choose 3) * 0.95^3 * 0.05^(3-3)
Since (3 choose 3) = 1, we have:
P(x=3) = 1 * 0.95^3 * 0.05
Finally, we can calculate the probability of at least 2 alarms ringing.
iii) Probability of at most 2 alarms ringing:
To calculate the probability that at most 2 alarms will ring, we need to find the probability of exactly 0 alarms ringing, plus the probability of exactly 1 alarm ringing, plus the probability of exactly 2 alarms ringing.
P(at most 2 alarms) = P(x=0) + P(x=1) + P(x=2)
The probability of exactly 0 alarms ringing is the complement of at least 1 alarm ringing, so:
P(x=0) = 1 - P(at least 1 alarm) = 1 - P(x=1) - P(x=2) - P(x=3)
We already know P(x=1), P(x=2), and P(x=3), so we can calculate P(x=0).
Finally, we can calculate the probability of at most 2 alarms ringing.
iv) Probability that all 3 alarms will ring:
We have already calculated this probability in ii), where P(x=3).
So, to summarize:
i) P(exactly 2 alarms) = P(x=2) = 3 * 0.95^2 * 0.05
ii) P(at least 2 alarms) = P(x=2) + P(x=3)
iii) P(at most 2 alarms) = P(x=0) + P(x=1) + P(x=2)
iv) P(all 3 alarms) = P(x=3) = 1 * 0.95^3 * 0.05
By substituting the given values, you can calculate the actual numerical probabilities for each case.