Probability
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Algebra II › Probability
A fair coin is tossed 50 times. What is the expected number of heads?
Explanation
The expected probably for the binomial distribution is n*p. N is the number of trials, in this case the 50 coin tosses. p is the probability of heads. Since the coin is a fair coin the probability is 
An experimental drug is created to reduce the amount of time patients feel sick with the common cold. In clinical trials of people suffering from the common cold, different participants taking the drug experienced symptoms for varying lengths of time. The scientists running the trial rounded each participant’s duration of symptoms to the nearest day, and used this information to develop the following probability distribution:
There were 
Explanation
If the probability distribution was constructed based on the real durations of participants’ symptoms, the 
Ann, Bob and Cathy are students working together on a group project for school. The project involves three tasks, each of which one of the three students will complete: creating a model, interviewing a local expert, and writing a report. No student has the time to complete more than one task, and all three of them have a strong preference for interviewing the local expert. They decide to randomly distribute the three tasks among themselves in a fair manner, such that all three of them have equal odds of completing their preferred task.
After some consideration, they reach for a coin. Bob flips it; if it lands heads, he creates the model; if tails, he interviews a local expert. After he is assigned a task by this method, Cathy flips a coin. If it lands heads, she writes the report; if tails, she is assigned the other remaining task. Ann is then assigned whichever task is left.
What is the probability that Ann is assigned to interview the expert?
0.25 or 25%
0.50 or 50%
1.0 or 100%
Cannot be determined
Explanation
Ann’s probability of interviewing the expert is reliant on two events: Bob’s coin toss and Cathy’s coin toss. For Ann to interview the expert, Bob must flip heads and Cathy must then flip heads as well. Each coin toss has a 50% odds of any one outcome, so Ann’s probability of interviewing the expert may be calculated as follows:
Event 1 * Event 2 = Ann’s probability
0.5 * 0.5 = 0.25
Therefore, Ann’s probability of interviewing the expert is 0.25, or 25%. Note that this method of allocating tasks is therefore not fair, as the three students will not have equal probabilities of interviewing the expert.
An experimental drug is created to reduce the amount of time patients feel sick with the common cold. In clinical trials of people suffering from the common cold, different participants taking the drug experienced symptoms for varying lengths of time. The scientists running the trial rounded each participant’s duration of symptoms to the nearest day, and used this information to develop the following probability distribution:
There were
Explanation
If the probability distribution was constructed based on the real durations of participants’ symptoms, the
A fair coin is tossed 50 times. What is the expected number of heads?
Explanation
The expected probably for the binomial distribution is n*p. N is the number of trials, in this case the 50 coin tosses. p is the probability of heads. Since the coin is a fair coin the probability is
Ann, Bob and Cathy are students working together on a group project for school. The project involves three tasks, each of which one of the three students will complete: creating a model, interviewing a local expert, and writing a report. No student has the time to complete more than one task, and all three of them have a strong preference for interviewing the local expert. They decide to randomly distribute the three tasks among themselves in a fair manner, such that all three of them have equal odds of completing their preferred task.
After some consideration, they reach for a coin. Bob flips it; if it lands heads, he creates the model; if tails, he interviews a local expert. After he is assigned a task by this method, Cathy flips a coin. If it lands heads, she writes the report; if tails, she is assigned the other remaining task. Ann is then assigned whichever task is left.
What is the probability that Ann is assigned to interview the expert?
0.25 or 25%
0.50 or 50%
1.0 or 100%
Cannot be determined
Explanation
Ann’s probability of interviewing the expert is reliant on two events: Bob’s coin toss and Cathy’s coin toss. For Ann to interview the expert, Bob must flip heads and Cathy must then flip heads as well. Each coin toss has a 50% odds of any one outcome, so Ann’s probability of interviewing the expert may be calculated as follows:
Event 1 * Event 2 = Ann’s probability
0.5 * 0.5 = 0.25
Therefore, Ann’s probability of interviewing the expert is 0.25, or 25%. Note that this method of allocating tasks is therefore not fair, as the three students will not have equal probabilities of interviewing the expert.
A spectator at a horse race is deciding which of three high-performing horses to bet on. Each horse’s ranks in past races can be expressed in terms of the following probability distributions:
The spectator wants to bet on the horse with the highest expected rank in its next race. Assuming that past performance is a good predictor of each horse’s performance in its respective next races, which horse should the spectator bet on?
Horse 1
Horse 2
Horse 3
Explanation
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.
Three students are playing a game with a fair six-sided die. If an even number is rolled, student A gets a point. If a number less than 4 is rolled, student B gets a point. If a prime number is rolled, student C gets a point. The die will be rolled fifteen times.
Is the game fair? In other words, do all three students have the same odds of getting a point?
No
Yes
Explanation
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.
Evaluate:
Explanation
Write the permutation formula.
Substitute the values in the given problem.
Cancel out the terms in the numerator and denominator.
The remaining terms in the numerator are:
The answer:
Evaluate:
Explanation
Write the permutation formula.
Substitute the values in the given problem.
Cancel out the terms in the numerator and denominator.
The remaining terms in the numerator are:
The answer: