A fair coin is tossed 5 times. If X is the discrete variable the # of tails obtained,
a) Draw the probability distribution .
b) Find P (X<5)
To solve this problem, we need to first understand the concept of a probability distribution and then determine the probability of getting less than 5 tails in 5 coin tosses.
a) Probability Distribution:
A probability distribution is a function that describes the likelihood of obtaining different outcomes in a specific event. In this case, we are interested in finding the probability distribution for the number of tails obtained when tossing a fair coin 5 times.
To draw the probability distribution, we need to find the probability of each possible outcome. In this case, the number of tails can range from 0 to 5, as there are 5 coin tosses.
Let's calculate the probabilities for each possible outcome:
Number of Tails (X) | Probability (P(X))
----------------------------------------
0 | ?
1 | ?
2 | ?
3 | ?
4 | ?
5 | ?
To find the probability for each outcome, we need to use the binomial probability formula:
P(X=k) = (n C k) * p^k * (1-p)^(n-k)
Where:
- n is the number of trials (coin tosses) = 5
- k is the number of successes (tails)
- p is the probability of success in one trial (heads or tails) = 0.5 for a fair coin toss
Using this formula, we can calculate the probabilities for each outcome:
Number of Tails (X) | Probability (P(X))
----------------------------------------
0 | (5 C 0) * (0.5)^0 * (0.5)^(5-0) = 1 * 1 * 0.03125 = 0.03125
1 | (5 C 1) * (0.5)^1 * (0.5)^(5-1) = 5 * 0.5 * 0.125 = 0.15625
2 | (5 C 2) * (0.5)^2 * (0.5)^(5-2) = 10 * 0.25 * 0.125 = 0.3125
3 | (5 C 3) * (0.5)^3 * (0.5)^(5-3) = 10 * 0.125 * 0.125 = 0.15625
4 | (5 C 4) * (0.5)^4 * (0.5)^(5-4) = 5 * 0.0625 * 0.5 = 0.15625
5 | (5 C 5) * (0.5)^5 * (0.5)^(5-5) = 1 * 0.03125 * 1 = 0.03125
Now we have the complete probability distribution:
Number of Tails (X) | Probability (P(X))
----------------------------------------
0 | 0.03125
1 | 0.15625
2 | 0.3125
3 | 0.15625
4 | 0.15625
5 | 0.03125
b) P(X < 5):
To find the probability of getting less than 5 tails in 5 coin tosses, we need to calculate the sum of the probabilities for getting 0, 1, 2, 3, and 4 tails.
P(X < 5) = P(X = 0) + P(X = 1) + P(X = 2) + P(X = 3) + P(X = 4)
P(X < 5) = 0.03125 + 0.15625 + 0.3125 + 0.15625 + 0.15625 = 0.8125
Therefore, the probability of getting less than 5 tails in 5 coin tosses is 0.8125.