If a coin is flipped 20 times, what is the probability of getting 5 heads?

Probability is a part of mathematics that deals with the possibility of happening of events. It is to forecast that what are the possible chances that the events will occur or the event will not occur. The probability as a number lies between 0 and 1 only and can also be written in the form of a percentage or fraction. The probability of likely event B is often written as P(B). Here P shows the possibility and B show the happening of an event. Similarly, the probability of any event is often written as P(). When the end outcome of an event is not confirmed we use the probabilities of certain outcomes—how likely they occur or what are the chances of their occurring.

Though probability started with a gamble, in the fields of Physical Sciences, Commerce, Biological Sciences, Medical Sciences, Weather Forecasting, etc., it has been used carefully.

To understand probability more accurately we take an example as rolling a dice:

The possible outcomes are — 1, 2, 3, 4, 5, and 6.

The probability of getting any of the outcomes is 1/6. As the possibility of happening of an event is an equally likely event so there are same chances of getting any number in this case it is either 1/6 or 50/3%.

Formula of Probability

Probability of an event = {Number of ways it can occur} ⁄ {Total number of outcomes}

P(A) = {Number of ways A occurs} ⁄ {Total number of outcomes}

Types of Events

• Equally Likely Events: After rolling dice the probability of getting any of the likely events is 1/6. As the event is an equally likely event so there is same possibility of getting any number in this case it is either 1/6 in fair dice rolling.
• Complementary Events: There is a possibility of only two outcomes which is an event will occur or not. Like a person will play or not play, buying a laptop or not buying a laptop, etc. are examples of complementary events.

If a coin is flipped 20 times, what is the probability of getting 5 heads?

Solution:

Use the binomial distribution directly. Let us assume that the number of heads is 

represented by x  (where an result of heads is regard as success) and in this case X = 5

Assuming that the coin is unbiased, you have a probability of success ‘p’

(where p is considered as success) is 1/2 and the probability of failure ‘q’ is 1/2

(where q is considered as failure). The number of trials is represented by the letter ’n’ and for this question n = 20.

Now just use the probability function for a binomial distribution:

P(X = x) = ⁿC_xp^xq^n-x

Using the information in the problem we get

P(X = 5) = (²⁰C₅)0.5⁵0.5¹⁵ 

= 0.014786

Similar Questions

Question 1: What is the probability of flipping a coin 20 times and getting 20 heads?

Answer:

Use the binomial distribution 

So suppose that the coin is balanced and has a head on one side of it and a tail on the other.

Then, P(head)=1⁄2

and 

P(tail)=1⁄2.

P(flipping a coin 20 times and getting 20 heads)

=(P(head))²⁰ = (1⁄2)²⁰

= 1⁄1048576

= 9.5367431640625×10^-7.

Question 2: A coin is tossed 20 times. What is the probability of getting at least 1 head?

Solution:

Use the binomial distribution 

Tossing a coin can give 2 outcomes.

So, tossing a coin 20 times can give (2^20) outcomes.

If we exclude the outcomes of getting at least one head; we will be left with the one and only option of getting all ‘tails’.

So, the probability of getting at least one head = [{(2²⁰) – 1} / (2²⁰)]

= [1 – {1 / (2²⁰)}].

= 0.999999