When a coin is tossed four times, sixteen equally likely outcomes are possible as shown below:
HHHH HHHT HHTH HHTT
HTHH HTHT HTTH HTTT
THHH THHT THTH THTT
TTHH TTHT TTTH TTTT
Let X denote the total number of tails obtained in the four tosses. Find the probability distribution of the random variable X. Leave your probabilities in fraction form.
To find the probability distribution of the random variable X, which denotes the total number of tails obtained in four coin tosses, we need to determine the probability of each possible value of X occurring.
In this case, X can take values from 0 to 4, because there can be 0, 1, 2, 3, or 4 tails in four coin tosses.
To find the probability of each value of X, we divide the number of outcomes that result in that value of X by the total number of equally likely outcomes.
Let's calculate the probabilities for each value of X:
X = 0 (no tails):
There is only one outcome that has no tails: HHHH
Therefore, the probability of X = 0 is 1/16.
X = 1 (one tail):
There are four outcomes that have one tail: HHHT, HHTH, HTHH, THHH
Therefore, the probability of X = 1 is 4/16 = 1/4.
X = 2 (two tails):
There are six outcomes that have two tails: HHTT, HTHT, HTTH, THHT, THTH, TTHH
Therefore, the probability of X = 2 is 6/16 = 3/8.
X = 3 (three tails):
There are four outcomes that have three tails: TTHH, TTHT, THTT, HTTT
Therefore, the probability of X = 3 is 4/16 = 1/4.
X = 4 (four tails):
There is only one outcome that has four tails: TTTT
Therefore, the probability of X = 4 is 1/16.
So, the probability distribution of the random variable X is as follows:
X = 0: Probability = 1/16
X = 1: Probability = 1/4
X = 2: Probability = 3/8
X = 3: Probability = 1/4
X = 4: Probability = 1/16