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.
Simulate the supply for 20 days using the random numbers below in order to calculate the expected average from 6,68,59,75,96,68,71,71,69,31,28,99,76,38,53,10,78,38,84,65
Well, that's quite a tall order! Let's break it down and have some fun with it, shall we?
To find the probability distribution of the random variable X, we need to determine the probability of each possible outcome. But first, let me grab my clown calculator...
Okay, let's get started with the possibilities. We have 16 outcomes in total.
Now, let's count how many tails are in each outcome:
- HHHH has 0 tails.
- HHHT has 1 tail.
- HHTH has 1 tail.
- HHTT has 2 tails.
- HTHH has 1 tail.
- HTHT has 2 tails.
- HTTH has 2 tails.
- HTTT has 3 tails.
- THHH has 1 tail.
- THHT has 2 tails.
- THTH has 2 tails.
- THTT has 3 tails.
- TTHH has 2 tails.
- TTHT has 3 tails.
- TTTH has 3 tails.
- TTTT has 4 tails.
Now that we know how many tails each outcome has, let's calculate their probabilities. Remember, we're looking for the probability distribution, so we need to find the probability of each number of tails (X) occurring.
- The probability of getting 0 tails is 1/16 (since HHHH is the only outcome with 0 tails).
- The probability of getting 1 tail is 3/16 (since there are three outcomes with 1 tail).
- The probability of getting 2 tails is 6/16 (since there are six outcomes with 2 tails).
- The probability of getting 3 tails is 4/16 (since there are four outcomes with 3 tails).
- The probability of getting 4 tails is 1/16 (since TTTT is the only outcome with 4 tails).
So, the probability distribution of the random variable X is:
X = 0, P(X = 0) = 1/16
X = 1, P(X = 1) = 3/16
X = 2, P(X = 2) = 6/16
X = 3, P(X = 3) = 4/16
X = 4, P(X = 4) = 1/16
And there you have it! The probability distribution of X in all its fractionally funny glory. Enjoy flipping those coins and embracing the randomness!
To find the probability distribution of the random variable X, which represents the total number of tails obtained in the four coin tosses, we need to determine the probability of each possible value of X.
In this case, X can be any integer value from 0 to 4, as there can be 0, 1, 2, 3, or 4 tails in four coin tosses.
To calculate the probability for each value of X, we count the number of outcomes that correspond to X tails and divide it by the total number of possible outcomes.
Let's go through each possible value of X and calculate its probability.
X = 0: There is only one outcome with no tails (TTTT). So, the probability of X = 0 is 1/16.
X = 1: There are four outcomes with exactly one tail (HTTT, THTT, TTHT, TTTH). So, the probability of X = 1 is 4/16 = 1/4.
X = 2: There are six outcomes with exactly two tails (HHTT, HTHT, HTTH, THHT, THTH, TTHH). So, the probability of X = 2 is 6/16 = 3/8.
X = 3: There are four outcomes with exactly three tails (HHTH, HTHH, THHH, TTHH). So, the probability of X = 3 is 4/16 = 1/4.
X = 4: There is only one outcome with four tails (HHHH). So, the probability of X = 4 is 1/16.
Therefore, the probability distribution of the random variable X is as follows:
X = 0: 1/16
X = 1: 1/4
X = 2: 3/8
X = 3: 1/4
X = 4: 1/16
These probabilities represent the likelihood of obtaining each possible value of X when the coin is tossed four times.
As you mentioned, there are 16 possible outcomes each with equal probability.
For X=0, there is only one case out of sixteen, namely HHHH. Therefore
X(0)=1/16.
For X=1, you will count the number of cases where T occurs only once. You should count 4 of such cases, therefore
X(1)=4/16=1/4
Repeat the calculation for X=2,3,4 and obtain the values of X(2), X(3), and X(4).
The sum of the values X(0) to X(5) should equal to 1.
If there are 16 possible outcomes for the 4 tosses, the probability of getting 4 tails = 1/16, 3 tails = 4/16 = 1/4, 2 tails = ?, 1 tail = ?, no tails = ?.
I'll let you calculate the remaining fractions.
As a check, the sum of these fractions must = 1.
I hope this helps.