How to find the Coin Toss Probability?

Probability is the branch of mathematics that deals with the occurrence of a particular event. It basically specifies how likely something is to happen. Example: What is the probability of raining in a clear sky? The range of probability lies between 0 and 1. The probability of an impossible event is 0 and the probability of a sure event is 1. The probability can also be expressed in terms of percentage. 

Terms related to probability

• Experiment: Any operation that provides well-defined outcomes is called experiment. For example: Tossing a coin or throwing a die is an experiment.
• Random Experiment: In any experiments, all possible outcomes but one does not know which exact outcome will occur. This is called Random experiment. For example: By tossing a coin, either heads or tails are obtained but one is not sure that only the head will occur or the tail will occur.
• Sample Space: Sample space is the set of all possible outcomes. Example: On throwing a coin we have 2 outcomes: heads and tails.
• Trial: It is a process by which the experiment is performed and the outcome is noted. For example: Picking up a card from a deck of 52 cards.
• Event: Each outcome of an experiment is called event. For example: Getting a head on throwing a coin is an event.
• Independent Events: When the occurrence of one event is not affected by the occurrence of another event then it is known as independent events. For Example, one can simultaneously flip a coin and throw a dice as they are separate events.
• Exhaustive Events: Two events are said to be exhaustive if their union is equal to sample space.
• Exclusive Events: When two events cannot occur at same time or the two events are disjoint, they are said to be exclusive events. For Example: On tossing a coin one can get either head or tail but not both.

Formula for probability

The formula for probability is,

Probability = Number of favorable outcomes/Total number of outcomes

For example:

P(A) = Number of ways A occur/Total number of outcomes

How to find the Coin Toss Probability?

It is known that a coin has two sides: Heads and Tails. It is not known which outcome will occur but one knows that there are 2 chances: one is head and the other is tail. It is a random experiment. Suppose there is an unbiased coin. So the total number of outcomes = 2 (Since 1 head and 1 tail) and one wants the head to occur. What is the probability of occurrence of head? Since 1 head is present in 1 coin and the total number of outcomes is 2

The probability of occurrence of head = 1/2

and the probability of occurrence of tail = 1/2

Finding probability for multiple coins

The probability of 1 coin is found but what about 2 coins or more than that. Let’s check it out. For 2 coins there are four net outcomes {HH, TH, HT, TT} since on first coin head or tail can occur. Similar is for second coin. So from the above case one wants to know the probability of finding,

• 2 heads
• 1 tail.

For first case,

Favourable outcomes = {HH} = 1

Probability of 2 heads is = 1/4

For second case,

Favorable outcomes = {TH, HT} = 2 (Here it is specifically mentioned that we have to find probability of 1 tail so 2 tails is not considered}

Probability of 1 tail = 2/4 = 1/2

But as the number of coins increases outcomes also increased so it’s not possible to find the favorable outcomes. Since tossing coins is independent event we use binomial distribution. The formula for binomial distribution is,

P(X) = ⁿC_x  × p^x × (1 – p)^{n – x}

Where n is total number of trials

x is a favorable trial, p is the probability of the favourable outcome. 

1 – p is the probability of unfavorable outcome. 

Let’s find the probability of 1 tail using 2 coins 

Probability of occurrence of tail = 0.5

Total number of trials = 2

Probability of head = 0.5

So the probability is = ²C₁ × (0.5) ¹ ×  (1 – 0.5) ¹ =0.5

Let us find the probability of 1 tail using 3 coins. Total trials is 3, Favorable trial is 1

Probability = ³C₁ × (0.5) ¹ × (1 – 0.5) ^{3 – 1} = {3/(1! 2! )} × 0.5 × 0.5² = 3 × (1/2) × (1/2) × (1/2) = 3/8 which is our answer. So the tossing of coins is not at all tough. Just remembering the formula and understanding the concept will help. 

Sample Problems

Question 1: A coin is tossed 3 times. What is the probability of getting same face on each coin? (Hint: The favorable events are HHH, TTT) 

Solution: 

The sample space are HTH, HHT, HHH, HTT, TTH, THT, TTT, THT

Total number of sample space events = 8

Favourable events = HHH, TTT

Number of favourable events = 2

Probability of getting same face on three coins = 2/8 = 1/4 = 0.25

Question 2: There are 10 coins, all are flipped at the same time. Find the probability of getting 5 heads. 

Solution:

Probability of getting head on 1 coin (p) = 0.50

Probability of getting tail on 1 coin (q) = 0.5

Number of coins (n) = 10

Number of heads(r) = 5

Using the formula of Binomial Distribution,

ⁿC_rp^rq^{n – r}

¹⁰C₅ (0.5) ⁵ (0.5)^{10 – 5} = [(10!) /(5! × 5!)] × (0.5)¹⁰ = 63/256 = 0.2461

Question 3: Two coins are tossed simultaneously. What is the probability of getting a head and a tail? 

Solution:

The sample space are TT, HH, TH, HT

Number of sample spaces = 2

Favourable events = HT, TH

Number of favourable events = 2

Probability of getting 1 head and 1 tail = 2/4 = 1/2 = 0.5