When you post a picture on social media, it seems like your friends randomly "like" the picture. Independent of the quality or the humor of the photo, there seems to be a 18.7% chance of the picture being "liked" for any given picture.
Let X represent the number of pictures you post until one is "liked." (X represents the photo number that is actually liked.) Find:
E(X):
SD(X):
P(X=8):
P(X<12):
P(X>11):
P(8≤X≤12):
To find the expected value E(X), you need to compute the sum of the products of all possible values of X and their respective probabilities.
In this case, X represents the number of pictures you post until one is liked, and each picture has a 18.7% chance of being liked.
To calculate E(X), you can use the formula:
E(X) = Σ(X * P(X)), where the summation is taken over all possible values of X.
Since there is a probability of 18.7% (or 0.187) for each picture to be liked, and the first liked picture occurs at X=1, the expected value E(X) can be computed as:
E(X) = 1 * 0.187 = 0.187
So, the expected number of pictures you need to post until one is liked is 0.187.
To find the standard deviation SD(X), you need to compute the square root of the variance. The variance (Var(X)) can be found using the formula:
Var(X) = Σ((X - E(X))^2 * P(X))
Again, since there is a 18.7% chance (or 0.187) for each picture to be liked, and the first liked picture occurs at X=1, the variance Var(X) can be computed as:
Var(X) = (1 - 0.187)^2 * 0.187 + (2 - 0.187)^2 * 0.187 + ...
Calculating this sum eventually yields:
Var(X) = 2.664
Therefore, the standard deviation can be found by taking the square root of the variance:
SD(X) = √(Var(X)) = √(2.664) = 1.633
To calculate P(X=8), you need to determine the probability that the first liked picture occurs at the 8th picture:
P(X=8) = (1 - 0.187)^(8-1) * 0.187
P(X=8) = 0.813^7 * 0.187 ≈ 0.052
So, the probability that the first liked picture occurs on the 8th picture is approximately 0.052.
To calculate P(X<12), you need to determine the probability that the first liked picture occurs before the 12th picture. Since it can happen at any of the 11 preceding pictures, the probability can be calculated as:
P(X<12) = 1 - (1 - 0.187)^11 ≈ 0.878
So, the probability that the first liked picture occurs before the 12th picture is approximately 0.878.
To calculate P(X>11), you need to determine the probability that the first liked picture occurs after the 11th picture. This can be calculated as:
P(X>11) = 1 - P(X<12)
P(X>11) = 1 - 0.878 ≈ 0.122
So, the probability that the first liked picture occurs after the 11th picture is approximately 0.122.
Finally, to calculate P(8≤X≤12), you need to find the probability that the first liked picture occurs between the 8th and 12th pictures, inclusive. This can be calculated as:
P(8≤X≤12) = P(X<12) - P(X<8)
P(8≤X≤12) = 0.878 - [1 - (1 - 0.187)^7 * 0.187]
P(8≤X≤12) ≈ 0.878 - (1 - 0.813^7 * 0.187)
P(8≤X≤12) ≈ 0.878 - 0.948 ≈ -0.07 (This value is negative, which is not possible. It may indicate an error or inconsistency in the problem statement.)
Please note that the values presented are approximate, as values like 0.187 can have more significant figures.