Probability Problems With Solved Examples

The probability of an event is how likely it is that the event will occur. Probability questions exist everywhere and at all times. Most importantly, probability lets us make predictions even though we cannot see into the unknowable future. Imagine a coin that you toss. There are two outcomes when throwing a coin: it lands on heads or tails. But there is an infinite number of possible outcomes—you can throw the coin many times and get a different answer every time. And yet we still talk about “heads” and “tails.

Basic probability concepts:

The field of probability theory deals with the analysis of random phenomena. There are two different types of random phenomena: events and experiments.

Experiments produce a list of outcomes, such as flipping a coin repeatedly to see what each side is or spinning a wheel to generate numbers between 0 and 36. Mathematically, experiments are usually described as the outcome of a single trial.

Events are those that have no outcome, such as rainfall. The probability of an event is solely dependent on the probability of the experiment from which it is derived and not from other events. For example, if you flip a coin 20 times and get 15 heads, you can safely assume that if you flip it again, you will get 15 heads with 95% confidence as each coin flip is independent of any other one.

Analysing probability questions:

There are two main methods to analyse probability questions-

First is the “trial method.” This is the most intuitive method. Here, the main strategy is to count the number of favourable outcomes and compare that to the total number of possible outcomes in the questions on probability theory.

So, if you’re calculating a probability like 15or 19and so on, this method would require counting all the numbers between 1 and 5 (except 1 and 2) to find which one(s) divide evenly by 5. Usually, the answer is obvious: 15 or 25etc.

For the second method, think about what’s going on and then guess which probabilities are involved. 

An Example of this method of solving probability questions: let’s say you’re told that a ski jumper has a 20% chance of winning and a 10% chance of placing second. What are the chances that he’ll end up on the podium? Well, 20% + 10% = 30%. It’s as simple as that.

Solved questions on probability:

Basic probability concepts can be understood easily by analyzing the following probability questions.

Question 1:

The probability of passing a mathematics test is 0.4. If a group of 10 students takes the test, what is the probability that 8 or 9 will pass it?

Solution:

In such type of problems, it is most convenient to work with “conditional probability”, i.e. P(A|B). In this case, P(A|B) = P(passing test | 8 or 9 will pass) = 10 × 0.4 × (1 – 0.4) = 20%.

Question 2:

A virus is present in 400 people. This virus causes acute fever with a temperature of 1000°C. One person randomly selected from this group tests positive and has a fever confirmed by a laboratory test. What is the probability that this person has the virus?

Solution:

Since 0 people don’t have the virus and one tested positive, we can say that 1% tested positive (this result will remain the same if we increase the number of individuals).

Question 3:

What probability will two heads and a tail be obtained if a coin is tossed three times?

Solution:

The probability of getting head on each toss is 12; therefore, we must multiply the three probabilities: 1/2×1/2×1/2=1/8 We can also write this as 8/32 or as 0.25. So, we got two correct results out of eight possible (heads and tails), i.e., 1/8 or 0.125, that is, 12.5%.

Question 4:

In a dice game, there are six faces with which the dice may fall, and a player can win 1 rupee each time he throws the die and gets 5. What is the probability of a player winning exactly 1 rupee in six dice throws?

Solution:

We know that a player can win 1 rupee on any of 6 faces, and he can win five coins in each throw; hence the probability of getting exactly one coin on each throw is P(1) = 6/6 Since we are interested in the probability of getting exactly one coin, then the total number of possible outcomes is (6)×(1) × 5/6 = 0.35.

Conclusion: 

Mathematicians define the probability of the occurrence of an event on a set of possible outcomes as “the ratio of the number of ways in which it can happen to all possible outcomes.” The basic probability concept is defined as the measure of the chance that an event will occur. A numerical value indicates the possibility of an event happening out of many possible outcomes. The values taken by probability are usually between 0 and 1, where 0 or 1 represent impossibility or certainty, respectively. In contrast, the value between these two extremes will evaluate how likely it is for an event to occur.