A fair coin is tossed three times and the events A, B, and C are defined as follows:
A:{ At least one head is observed }
B:{ At least two heads are observed }
C:{ The number of heads observed is odd }
Find the following probability by summing the probabilities of the appropriate sample points (note that 0 is an even number):
P((not A) or (not B) or C) =
Ah, probabilities and coin tosses, always a fun topic! Let's break down this question and find the desired probability.
First, let's calculate the probability of not event A occurring. Event A is defined as "at least one head is observed" in three coin tosses. So, the probability of not getting any heads in three tosses is (1/2) * (1/2) * (1/2) = 1/8.
Next, let's calculate the probability of not event B occurring. Event B is defined as "at least two heads are observed" in three coin tosses. So, the probability of getting no heads or just one head is (1/2) * (1/2) * (1/2) + 3*(1/2) * (1/2) * (1/2) = 1/8 + 3/8 = 4/8 = 1/2.
Lastly, let's calculate the probability of event C occurring. Event C is defined as "the number of heads observed is odd" in three coin tosses. Out of the eight possible outcomes, four of them have an odd number of heads (HTT, THT, TTH, HHH). So, the probability of event C occurring is 4/8 = 1/2.
Now, to find the probability of the entire expression, we need to calculate the probability of (not A) or (not B) or C. This is equivalent to 1 - P(A and B and not C).
The probability of A and B and not C occurring is the probability of getting exactly two heads (HHT, HTH, THH) which is 3/8.
So, the probability of (not A) or (not B) or C occurring is 1 - (3/8) = 5/8.
Therefore, P((not A) or (not B) or C) = 5/8.
So, in conclusion, the probability of the expression is 5/8. Just remember, probabilities are just like jokes - sometimes they're odd, but they can always make you laugh!
To find the probability of the event (not A) or (not B) or C, we need to consider the sum of the probabilities of the sample points that satisfy each condition.
Let's calculate each probability separately and then sum them up.
1. P(not A): This event occurs when no heads (i.e., all tails) are observed.
The sample points that satisfy this condition are (TTT) only.
Since there is only one sample point out of a total of 2^3 = 8 possible outcomes, the probability is 1/8.
2. P(not B): This event occurs when exactly zero or one head is observed.
The sample points that satisfy this condition are (TTT, TTH, THT, HTT, THH, HTH, HHT).
There are a total of 7 sample points satisfying this condition, so the probability is 7/8.
3. P(C): This event occurs when the number of heads observed is odd.
The sample points that satisfy this condition are (TTT, TTH, THT, HTT, HTH, THH, HHH).
Out of 8 sample points, there are 7 that satisfy this condition, so the probability is 7/8.
Now, to find the probability of (not A) or (not B) or C, we sum up the probabilities of each event:
P((not A) or (not B) or C) = P(not A) + P(not B) + P(C)
= 1/8 + 7/8 + 7/8
= 15/8
= 1.875
Therefore, the probability of the event (not A) or (not B) or C is 1.875 or 15/8.
To find the probability of the event P((not A) or (not B) or C), we need to sum the probabilities of the sample points that satisfy this event.
Let's break it down into the three conditions: (not A), (not B), and C.
1. (not A): This means that no heads are observed. The only sample point that satisfies this condition is TTT (tails, tails, tails). The probability of getting TTT is (1/2) * (1/2) * (1/2) = 1/8.
2. (not B): This means that at most one head is observed. The sample points that satisfy this condition are TTT, HTT, THT, and TTH (where H represents heads and T represents tails). The probability of getting each of these sample points is 1/8.
3. C: This means that the number of heads observed is odd. The sample points that satisfy this condition are HTT, THT, and TTH. The probability of getting each of these sample points is 1/8.
Now, we can sum up the probabilities of these sample points:
P((not A) or (not B) or C) = P(not A) + P(not B) + P(C)
= 1/8 + 1/8 + 1/8
= 3/8
Therefore, the probability of the event P((not A) or (not B) or C) is 3/8.