When a coin is tossed 100 times, the probability of obtaining at most

Tammy Fisher

Tammy Fisher

Answered question

2021-11-12

If a coin is flipped 100 times, the probability of obtaining 49 or fewer heads is 0.4602 and the probability of obtaining 50 heads is .0796 
(a) what is the probability of obtaining more than 50 heads? 
(b) what is the probability of obtaining at most 50 heads?

Answer & Explanation

Robert Harris

Robert Harris

Beginner2021-11-13Added 23 answers

Step 1
Given information:
It is given that a coin is flipped for 100 times. That is, n=100. It is known that all the outputs of 100 flips will be independent of each other, imp-lying independence among each other. As a result, the 100 flips can be considered as 100 independent trials.
Consider the event that the outcome is head as a “success”. Consider the probability of getting head as p, that is, the probability of success in each trial is p. Then q=1p.
Consider X as the number of heads. Then, X has a Binomial distribution with parameters n=100,p.
The pdf of X is given below:
P(x)={(100x)(p)x(1p)100x,x=0, 1, , 100; 0

Step 2 a) Obtain the probability of obtaining more than 50 heads.
It is given that the probability of obtaining at most 49 heads is PX49=0.4602.
The probability of obtaining 50 heads is PX=50=0.0796.
The probability of obtaining more than 50 heads is obtained as 0.4602 from the calculation given below:
PX>50=1-PX50
=1-PX49+PX=50
=1-0.4602+0.0796
=1-0.5398
=0.4602
Thus, the probability of obtaining more than 50 heads is 0.4602.
Step 3
b) Obtain the probability of obtaining at most 50 heads.
The probability of obtaining at most 50 heads is obtained as 0.5398 from the calculation given below:
PX50=PX49+PX=50
=0.4602+0.0796
=0.5398
Thus, the probability of obtaining at most 50 heads is 0.5398.

xleb123

xleb123

Skilled2023-05-26Added 181 answers

(a) The probability of obtaining more than 50 heads can be calculated by subtracting the sum of the probabilities of obtaining 50 or fewer heads from 1. Mathematically, this can be expressed as:
P(More than 50 heads)=1P(At most 50 heads)
(b) The probability of obtaining at most 50 heads can be calculated by summing the probabilities of obtaining 50 heads and 49 or fewer heads. Mathematically, this can be expressed as:
P(At most 50 heads)=P(50 heads)+P(49 or fewer heads)
Given that the probability of obtaining 49 or fewer heads is 0.4602 and the probability of obtaining 50 heads is 0.0796, we can substitute these values into the equations:
(a) P(More than 50 heads)=10.4602=0.5398
(b) P(At most 50 heads)=0.0796+0.4602=0.5398
Therefore, the probability of obtaining more than 50 heads is 0.5398, and the probability of obtaining at most 50 heads is also 0.5398.
Andre BalkonE

Andre BalkonE

Skilled2023-05-26Added 110 answers

Result:
(a) The probability of obtaining more than 50 heads is 0.5398.
(b) The probability of obtaining at most 50 heads is 0.5398.
Solution:
To solve the given problem, let's define the random variable X as the number of heads obtained when a coin is flipped 100 times. We are given the following probabilities:
P(X49)=0.4602
P(X=50)=0.0796
We need to find:
(a) The probability of obtaining more than 50 heads: P(X>50)
(b) The probability of obtaining at most 50 heads: P(X50)
(a) To find the probability of obtaining more than 50 heads, we can use the complement rule. The complement of obtaining more than 50 heads is obtaining 50 or fewer heads. Therefore:
P(X>50)=1P(X50)
Substituting the given values:
P(X>50)=10.4602
Therefore, the probability of obtaining more than 50 heads is 0.5398.
(b) To find the probability of obtaining at most 50 heads, we can use the cumulative probability up to 50 heads. Therefore:
P(X50)=P(X49)+P(X=50)
Substituting the given values:
P(X50)=0.4602+0.0796
Therefore, the probability of obtaining at most 50 heads is 0.5398.

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