Question

When ten fair coins are flipped, the event of interest is the number of heads. Let this number be a random variable. (a) Find the pmf of this random variable. (b) What is the probability that the random variable is between six and nine inclusive? (c) What is the probability that the random variable is greater than or equal to eight?

Answer 1

The pmf of the random variable representing the number of heads when ten fair coins are flipped can be calculated using the binomial distribution formula. The probabilities for each possible outcome ranging from 0 to 10 heads can be determined. The probability that the random variable is between six and nine inclusive can be found by summing the probabilities of getting 6, 7, 8, and 9 heads. The probability that the random variable is greater than or equal to eight can be found by summing the probabilities of getting 8, 9, and 10 heads.

Step-by-step explanation:

To find the pmf (probability mass function) of the random variable, we need to determine the probability of each possible outcome. In this case, the number of heads can range from 0 to 10, so we need to find the probability of getting 0, 1, 2, ..., 10 heads. Since each coin flip is independent and has a 50% chance of resulting in heads, we can use the binomial distribution formula to calculate the probabilities.

1. p(0 heads) = (10 choose 0) × (0.5)⁰ × (0.5)
   `^{(10-0)` = 0.000976562
2. p(1 head) = (10 choose 1) × (0.5)¹ × (0.5)
   `^{(10-1)` = 0.009765625
3. p(2 heads) = (10 choose 2) × (0.5)² × (0.5)
   `^{(10-2)` = 0.0439453125
4. ...
5. p(10 heads) = (10 choose 10) × (0.5)¹⁰ × (0.5)
   `^{(10-10)` = 0.000976562

So the pmf of the random variable is:
0: 0.000976562
1: 0.009765625
2: 0.0439453125
...
10: 0.000976562

(b) To find the probability that the random variable is between six and nine inclusive, we need to sum up the probabilities of getting 6, 7, 8, and 9 heads:
p(6 ≤ X ≤ 9) = p(6) + p(7) + p(8) + p(9) = 0.205078125

(c) To find the probability that the random variable is greater than or equal to eight, we need to sum up the probabilities of getting 8, 9, and 10 heads:
p(X ≥ 8) = p(8) + p(9) + p(10) = 0.1064453125

Answer 2

The random variable representing the number of heads when ten fair coins are flipped follows a binomial distribution, with the pmf calculated using combinations and powers of the probability of a single coin landing heads. To find specific probabilities, apply the pmf formula to each event of interest and sum the relevant probabilities.

Step-by-step explanation:

When ten fair coins are flipped, the event of interest is the number of heads, which is a random variable, often denoted as X. The probability mass function (pmf) of X follows a binomial distribution since each coin has two possible outcomes (heads or tails) and each flip is independent of the others.

To find the pmf of X:

• The number of trials (n) is 10.
• The probability of success (getting a head on a single flip), denoted as p, is 0.5.

The pmf can be calculated using the formula:

P(X = k) = C(n, k) × p^k × (1-p)^(n-k)

where C(n, k) is the combination of n items taken k at a time.

(b) The probability that the random variable is between six and nine inclusive is:

P(6 ≤ X ≤ 9) = P(X = 6) + P(X = 7) + P(X = 8) + P(X = 9). Each of these can be calculated using the pmf formula.

(c) The probability that the random variable is greater than or equal to eight is:

P(X ≥ 8) = P(X = 8) + P(X = 9) + P(X = 10). Again, use the pmf formula for each of these probabilities.