To answer this question, you could either use the binomial theorem or multiply . Both are detailed below.

Note: If you need help understanding Sigma Notation, please visit: https://www.varsitytutors.com/hotmath/hotmath_help/topics/sigma-notation-of-a-series; additionally, the notation  is often read out loud as “n choose k” and is another way to write For more information on combinations, visit https://www.varsitytutors.com/hotmath/hotmath_help/topics/combinations

The Binomial Theorem is:

In our case, n=4. Plugging this in, we get:

Alternatively, we can solve this problem by multiplying to get the following:

The expected value of each horse’s rank can be calculated as follows:

Expected value:

Horse 1:

Horse 2:

Horse 3:

The expected value predicts each horse’s rank in the upcoming race. Horse 1’s probability distribution yields the lowest expected value (i.e., the closest value to first place, represented by 1), so Horse 1 can be expected to rank most highly.

Expected value identifies the most likely outcome, but other outcomes may still result, including unlikely outcomes. Although the horse the spectator bet on had the highest likelihood among the other horses of ranking well in its next race, the race was still ultimately subject to chance. Additionally, the three horses’ expected ranks differed only slightly--2.1, 2.5 and 2.6--meaning that the horse with the highest expected rank had only a slightly greater likelihood of performing more highly than the other two horses to begin with. As a result, chance was a relatively significant determinant of outcomes.

The sample space for the die roll is {1, 2, 3, 4, 5, 6}. Student A gets a point if 2, 4 or 6 is rolled. Student B gets a point if 1, 2 or 3 is rolled. Student C gets a point if 2, 3 or 5 is rolled. Therefore, all three students have a 3/6 = ½ = 0/5 = 50% chance of getting a point on each die roll. The number of times the die is rolled does not affect each student’s probability of getting a point; the probability is the same in each roll so long as neither the die nor the point rules change. 

If the probability distribution was constructed based on the real durations of participants’ symptoms, the  probability corresponding to a -day duration of symptoms represents that  of the total number of trial participants experienced symptoms for about  days. There were  participants, so the number of participants who experienced symptoms for about  days must have been  participants.

The expected duration of a randomly selected patient’s symptoms can be calculated using the expected value yielded by this probability distribution. This expected value can be calculated as follows:

Expected value:

Therefore, the most likely duration of symptoms a randomly selected participant would have is  days.

 As long as the coin is truly fair, meaning that there is a 50% chance that it will land on heads and a 50% chance that it will land on tails, the coin will still be fair on the tenth flip. Past results do not affect the odds of a particular result on a subsequent identical event. Therefore, flipping this coin would be a fair way to decide which of the two students will get to keep the five-dollar bill, since each one of them would have a 50% probability of keeping the bill.