Given P(A) = m is the probability of event A occurring in any given trial:
1. State the range of possible values of m
2. Suppose two trials are performed independently. Find, in terms of m, the probability of A occurring:
a) Exactly once
b) At least once
I am very confused of how to solve this because it only involves variables with no numbers.
P = 0, never happens
P = 1, always happens
binomial probability
P(n,r) = C(n,r) m^r (1-m)^(n-r)
for exactly r = 1 for n = 2
C(2,1) = 2!/[1!(2-1)!] = 2
P(2,1)= 2 m^1 (1-m)^1 = 2 m(1-m)
in other word yes-no or no-yes :)
now for twice
P(2,2) = 1 m^2 (m-1)^0 = m^2
which we all know anyway
so for either
P(2,2)+P(2,1) = m^2+2m(1-m) =2m-m^2
1. Ah, the range of possible values for m, you ask? Well, my friend, since probabilities lie between 0 and 1 (inclusive), the possible values of m would also fall within that lovely range. So, 0 ≤ m ≤ 1. Isn't that neat?
2. Now, let's tackle the probability of event A occurring in two trials:
a) The probability of event A occurring exactly once in two independent trials is a bit of a fun calculation! It's like finding a needle in a haystack, or in this case, two haystacks. The probability can be expressed as 2m(1 - m). Clever, isn't it?
b) Ah, the joy of "at least once"! To find the probability of A occurring at least once in two independent trials, we can use a little trick. It's a playful game of subtracting the probability of A not occurring in both trials from the total probability. So, the probability can be expressed as 1 - (1 - m)².
I hope that helps add a little humor and clarity to your confusion!
1. The range of possible values of m is between 0 and 1, inclusive. This means that m can take any value between 0 (indicating zero probability) and 1 (indicating certainty or a probability of 100%).
2. For two independent trials:
a) The probability of A occurring exactly once can be calculated by considering the cases where A occurs in the first trial and not in the second trial, and when A does not occur in the first trial and occurs in the second trial. The probability of A occurring exactly once is given by the sum of these probabilities:
P(A occurs exactly once) = P(A occurs in the first trial and not in the second trial) + P(A does not occur in the first trial and occurs in the second trial)
= P(A) * (1 - P(A)) + (1 - P(A)) * P(A)
Simplifying, we get:
P(A occurs exactly once) = 2 * P(A) * (1 - P(A))
b) The probability of A occurring at least once can be calculated by considering the cases where A occurs in the first trial, A occurs in the second trial, and when A occurs in both trials. The probability of A occurring at least once is given by the sum of these probabilities:
P(A occurs at least once) = P(A occurs in the first trial) + P(A occurs in the second trial) - P(A occurs in both trials)
= P(A) + P(A) - P(A) * P(A)
Simplifying, we get:
P(A occurs at least once) = 2 * P(A) - P(A)^2
To state the range of possible values of m, we need to consider the properties of probability. The probability of any event ranges from 0 to 1, inclusive. So, the range of possible values for m is 0 ≤ m ≤ 1.
Now let's move on to calculating probabilities in terms of m for two independent trials:
a) To find the probability of event A occurring exactly once in two independent trials, we need to consider the two possible cases: A occurring in the first trial and not in the second, or A not occurring in the first trial and occurring in the second.
Case 1: A occurs in the first trial and not in the second trial.
The probability of A occurring in the first trial is m. The probability of A not occurring in the second trial is 1 - m. So, the combined probability for this case is m * (1 - m).
Case 2: A does not occur in the first trial and occurs in the second trial.
The probability of A not occurring in the first trial is 1 - m. The probability of A occurring in the second trial is m. So, the combined probability for this case is (1 - m) * m.
Since these two cases are mutually exclusive (they cannot occur simultaneously), we can add their probabilities to get the total probability of A occurring exactly once:
P(A occurs exactly once) = m * (1 - m) + (1 - m) * m = 2m(1 - m)
b) To find the probability of event A occurring at least once in two independent trials, we can approach it using the complement rule. The complement of "A occurs at least once" is "A does not occur in either trial."
The probability of A not occurring in a single trial is 1 - m. Since the two trials are independent, the probability of A not occurring in either trial is the product of these individual probabilities. Therefore, the probability of A occurring at least once is the complement of this probability:
P(A occurs at least once) = 1 - (1 - m) * (1 - m) = 1 - (1 - m)^2 = 1 - (1 - 2m + m^2) = 1 - 1 + 2m - m^2 = 2m - m^2
So, in terms of m, the probability of A occurring exactly once is 2m(1 - m), and the probability of A occurring at least once is 2m - m^2.