an experiment consists of 4 tosses of a coin.denoting the outcomes HHTH,THTT,..and assuming that all 16 outcomes are equally likely, find the probability distribution for the total number of heads.
Denote all the outcomes. Count the number of of outcomes with 0 heads, 1 head, 2 heads, 3 heads and 4 heads. Graph the probability distribution from that data.
To find the probability distribution for the total number of heads, we need to determine the number of heads for each possible outcome and calculate their probabilities.
In this experiment, we have 4 tosses of a coin, so the total number of heads can range from 0 to 4.
Let's go through each possible outcome and count the number of heads:
1. HHTH: This outcome has 3 heads.
2. THTT: This outcome has 1 head.
3. HHHT: This outcome has 3 heads.
4. HTTH: This outcome has 2 heads.
5. TTHH: This outcome has 2 heads.
6. HTHT: This outcome has 2 heads.
7. THHT: This outcome has 2 heads.
8. TTTT: This outcome has 0 heads.
9. HHHH: This outcome has 4 heads.
10. HTHH: This outcome has 3 heads.
11. THHH: This outcome has 3 heads.
12. HHHT: This outcome has 3 heads.
13. HHTT: This outcome has 2 heads.
14. HTTH: This outcome has 2 heads.
15. HHTT: This outcome has 2 heads.
16. HTHH: This outcome has 3 heads.
Now, let's calculate the probabilities for each count of heads. Since we assumed that all 16 outcomes are equally likely, we need to determine how many outcomes have each count of heads and divide it by the total number of outcomes (16).
Number of heads: 0
Number of outcomes: 1 (TTTT)
Probability: 1/16
Number of heads: 1
Number of outcomes: 1 (THTT)
Probability: 1/16
Number of heads: 2
Number of outcomes: 4 (HTTH, TTHH, HTHT, THHT)
Probability: 4/16 = 1/4
Number of heads: 3
Number of outcomes: 6 (HHTH, HHHT, HTHH, HHHT, HHTT, HTHH)
Probability: 6/16 = 3/8
Number of heads: 4
Number of outcomes: 4 (HHHH)
Probability: 4/16 = 1/4
Therefore, the probability distribution for the total number of heads is as follows:
Number of heads: 0, Probability: 1/16
Number of heads: 1, Probability: 1/16
Number of heads: 2, Probability: 1/4
Number of heads: 3, Probability: 3/8
Number of heads: 4, Probability: 1/4
To find the probability distribution for the total number of heads, we need to determine the probability of getting each possible number of heads from 0 to 4 in the 4 tosses of the coin.
First, let's count the number of outcomes that will result in each possible number of heads:
- 0 Heads: The outcomes that result in 0 heads are TTTT, so there is only one outcome.
- 1 Head: The outcomes that result in 1 head are HTTT, THTT, TTHT, and TTTH, so there are 4 outcomes.
- 2 Heads: The outcomes that result in 2 heads are HHTT, HTHT, HTTH, THHT, TTHH, and THTH, so there are 6 outcomes.
- 3 Heads: The outcomes that result in 3 heads are HTHH, HHTH, HHHT, and THHH, so there are 4 outcomes.
- 4 Heads: The outcome that results in 4 heads is HHHH, so there is only one outcome.
Now, to find the probability for each number of heads, we need to divide the number of outcomes for each number of heads by the total number of outcomes, which is 16:
- Probability of 0 Heads: 1/16
- Probability of 1 Head: 4/16 = 1/4
- Probability of 2 Heads: 6/16 = 3/8
- Probability of 3 Heads: 4/16 = 1/4
- Probability of 4 Heads: 1/16
Therefore, the probability distribution for the total number of heads in the 4 tosses of the coin is:
Number of Heads: 0 1 2 3 4
Probability: 1/16 1/4 3/8 1/4 1/16