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Answer:
Step-by-step explanation:
Hello!
Distribution in attachment.
The variable of interest is:
X: Number of successes after flipping a coin 5 times.
If you check the binomial criteria:
The number of trials is fixed: n=5
There are only two possible outcomes "success" or "failure"
Each flip of the coin is independent of the others.
The probability of success in the same from one trial to another, in this case, if we consider the coin to be balanced, the probability of success is p=0.5
The histogram shows the probability of obtaining X number of success in 5ve flips of a coin (y-axis) vs the number of successes counted each (x-axis)
Statements:
1) The bar for any number k represents the probability of getting k successes in 5 flips. Correct. The histogram shows the probability of obtaining X number of success in 5ve flips of a coin (y-axis) vs the number of successes counted each (x-axis). Each bar represents the probability of success for each possible value.
2) The number of successes, k, can range from 0 (no success) to 5 (all successes). Correct. The variable count the number of successes after flipping a coin 5 times. It can happen that you flip it and all the flips turn to be failures (X=0), that you flip it 5 times and only one turns out to be a success and the other 4 are failures (X=1), and so on until you flip it 5 times and all flips are successes (X=5)
3) Each coin flip is independent; it is not affected by any other coin flip. Correct, if not, this variable wouldn't have a binomial distribution as specified in the text.
4) For 5 coin flips, P(2 heads) = P(3 heads). Correct
Looking at the histogram, the bars for "2 successes" and "3 successes" have the same height, a little above 0.3, this means that both values have the same probability of occurrence.
5) The sum of the probabilities shown in the binomial distribution is p.
Incorrect.
For the binomial distribution "p" represents the probability of success for each trial, in this case, flipping the coin once.
For this distribution, as well as for other probability distribution, the sum of all probabilities is always 1, if not, then it is not a probability distribution.
I hope this helps!
