There are two different types of probability that we often talk about: theoretical probability and experimental probability.
Theoretical probability describes how likely an event is to occur. We know that a coin is equally likely to land heads or tails, so the theoretical probability of getting heads is 1/2.
Experimental probability describes how frequently an event actually occurred in an experiment. So if you tossed a coin 20 times and got heads 8 times, the experimental probability of getting heads would be 8/20, which is the same as 2/5, or 0.4, or 40%.
The theoretical probability of an event will always be the same, but the experimental probability is affected by chance, so it can be different for different experiments. The more trials you carry out (for example, the more times you toss the coin), the closer the experimental probability is likely to be to the theoretical probability.
Maybe you could try tossing a coin 20 times to see how close your experimental probability is to the theoretical probability.
Student: I just flipped a coin several times and the coin landed on heads more than tails. I had 9 heads and only 6 tails. I don't understand why that happened since heads and tails should be equally likely.
Mentor: How do you know that landing on heads is just as likely as landing on tails when a coin is tossed?
Student: Well, a coin only has two sides (heads and tails) so that means that flipping a coin can have two possible outcomes. The chances for both are equal since the coin is essentially the same on both sides. Therefore, the chances of a coin landing on heads would be 1/2, and the chances of it landing on tails would also be 1/2.
Mentor: That is right! What you just found was the theoretical probability of a coin landing on heads (1/2 or 50%) and a coin landing on tails (1/2 or 50%).
Student: Alright, well why aren't my results from flipping a coin just now the same as the theoretical probability of flipping a coin?
Mentor: Theoretical probability is a way of estimating what could happen based on the information that you have; it is a calculation. Theoretical probability cannot predict what the actual results will be, but it does give you an idea of what is likely to happen in a situation.
Student: I understand that. So the results of flipping a coin should be somewhere around 50% heads and 50% tails since that is the theoretical probability.
Mentor: Yes! Now let's look at the coin flipping game that you just played. What were the results?
Student: The coin landed on heads 9 times and on tails 6 times. That means I flipped the coin 15 times.
Mentor: OK, we are going to use this information to find another form of probability called experimental probability. To find the experimental probability, you find the ratio of the number of trials with a certain outcome to total number of trials. Experimental probability of winning= # of trials with a certain outcome/# of total trials. So let's first find the experimental probability of flipping heads. For this situation the number of games won would be the number of flips that landed on heads. That would be 9.
Student: The number of games played was 15, so that means that the experimental probability is 9/15 (or simplified, 3/5)!
Mentor: Now what would the experimental probability of flipping tails be?
Student: Well, the number of games won in this situation would be the number of times that I flipped tails, so 6. Then, I played 15 games so the ratio would be 6/15 (or simplified, 2/5).
Mentor: Great. Now, can you write these experimental probabilities as percents?
Student: I would multiply 3/5 by 100% and get 60% as the experimental probability of flipping heads. Then for tails I would multiply 2/5 by 100% and get 40%. The experimental probability of flipping tails is 40%.
Mentor: The experimental probabilities were 40% tails and 60% heads. This does not precisely match with the theoretical probability of 50% tails and 50% heads. However, they are not too far off. Let's do an experiment! Using the coin toss activity, toss the coin 25 times and then 150 times.
Student: OK, after 25 tosses I got 11 heads and 14 tails, and after 150 tosses I got 71 heads and 79 tails.
Mentor: Alright, we know the theoretical probability will be 50% heads and 50% tails no matter how many trials, but what would the experimental probability be in this experiment?
Student: For 25 tosses the probability of heads would be 11/25 (44%) and for tails would be 14/25 (56%). For 125 tosses the probability of heads would be 71/150 (about 47%) and the probability of tails would be 79/150 (about 53%).
Mentor: Now which results have the experimental probability closer to the theoretical probability?
Student: After 25 tosses, the experimental probabilities of heads and tails are not very close to 50%. However, after 150 tosses the experimental probabilities for heads and tails are much closer to 50%.
Mentor: Can you make an educated guess at what that means?
Student: Well, it seems that with more tosses, the resulting experimental probabilities are closer to the theoretical probabilities.
Mentor: Good job! As the amount of trials (in this case a trial is flipping a coin) increases, the experimental probability gets closer to the theoretical probability. You can test this concept with the Crazy Choices Game.
Problems on coin toss probability are explained here with different examples.When we flip a coin there is always a probability to get a head or a tail is 50 percent.
Suppose a coin tossed then we get two possible outcomes either a ‘head’ (H) or a ‘tail’ (T), and it is impossible to predict whether the result of a toss will be a ‘head’ or ‘tail’.
The probability for equally likely outcomes in an event is:
Number of favourable outcomes ÷ Total number of possible outcomes
Total number of possible outcomes = 2
(i) If the favourable outcome is head (H).
Number of favourable outcomes = 1.
Therefore, P(getting a head)
= P(H) = total number of possible outcomes
= 1/2.
(ii) If the favourable outcome is tail (T).
Number of favourable outcomes = 1.
Therefore, P(getting a tail)
Number of favorable outcomes= P(T) = total number of possible outcomes
= 1/2.
Word Problems on Coin Toss Probability:
1. A coin is tossed twice at random. What is the probability of getting
(i) at least one head
(ii) the same face?
Solution:
The possible outcomes are HH, HT, TH, TT.
So, total number of outcomes = 4.
(i) Number of favourable outcomes for event E
= Number of outcomes having at least one head
= 3 (as HH, HT, TH are having at least one head).
So, by definition, P(F) = \(\frac{3}{4}\).
(ii) Number of favourable outcomes for event E
= Number of outcomes having the same face
= 2 (as HH, TT are have the same face).
So, by definition, P(F) = \(\frac{2}{4}\) = \(\frac{1}{2}\).
2. If three fair coins are tossed randomly 175 times and it is found that three heads appeared 21 times, two heads appeared 56 times, one head appeared 63 times and zero head appeared 35 times.
What is the probability of getting
(i) three heads, (ii) two heads, (iii) one head, (iv) 0 head.
Solution:
Total number of trials = 175.
Number of times three heads appeared = 21.
Number of times two heads appeared = 56.
Number of times one head appeared = 63.
Number of times zero head appeared = 35.
Let E1, E2, E3 and E4 be the events of getting three heads, two heads, one head and zero head respectively.(i) P(getting three heads)
Number of times three heads appeared= P(E1) = total number of trials
= 21/175
= 0.12
(ii) P(getting two heads)
Number of times two heads appeared= P(E2) = total number of trials
= 56/175
=
0.32
(iii) P(getting one head)
Number of times one head appeared= P(E3) = total number of trials
= 63/175
= 0.36
(iv) P(getting zero head)
Number of times zero head appeared= P(E4) = total number of trials
= 35/175
= 0.20
Note: Remember when 3 coins are tossed randomly, the only possible outcomes
are E2, E3, E4 andP(E1) + P(E2) + P(E3) + P(E4)
= (0.12 + 0.32 + 0.36 + 0.20)
= 1
3. Two coins are tossed randomly 120 times and it is found that two tails appeared 60 times, one tail appeared 48 times and no tail appeared 12 times.
If two coins are tossed at random, what is the probability of getting
(i) 2 tails,
(ii) 1 tail,
(iii) 0 tail
Solution:
Total number of
trials = 120
Number of times 2 tails appear = 60
Number of times 1 tail appears
= 48
Number of times 0 tail appears
= 12
(i) P(getting 2 tails)
Number of times 2 tails appear= P(E1) = total number of trials
= 60/120
= 0.50
(ii) P(getting 1 tail)
Number of times 1 tail appear= P(E2) = total number of trials
= 48/120
= 0.40
(iii) P(getting 0 tail)
Number of times no tail appear= P(E3) = total number of trials
= 12/120
= 0.10
Note:
Remember while tossing 2 coins simultaneously, the only possible outcomes are E1, E2, E3 and,P(E1) + P(E2) + P(E3)
= (0.50 + 0.40 + 0.10)
= 1
4. Suppose a fair coin is randomly
tossed for 75 times and it is found that head turns up 45
times and tail 30 times. What is the probability of getting (i)
a head and (ii) a tail?
Solution:
Total number of trials = 75.
Number of times head turns up = 45
Number of times tail turns up = 30
(i) Let X be the event of getting a head.
P(getting a head)
Number of times head turns up= P(X) = total number of trials
= 45/75
= 0.60
(ii) Let Y be the event of getting a tail.
P(getting a tail)
Number of times tail turns up= P(Y) = total number of trials
= 30/75
= 0.40
Note: Remember when a fair coin is tossed and then X and Y are the only possible outcomes, and
P(X) + P(Y)
= (0.60 + 0.40)
= 1
Moving forward to the theoretical probability which is also known as classical probability or priori probability we will first discuss about collecting all possible outcomes and equally likely outcome. When an experiment is done at random we can collect all possible outcomes
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