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In this week’s class with News18, we are discussing mathematics and more particularly a lesson in probability. This seemingly simple chapter can have great significance in your life. It can help you understand your chances at winning a lottery or help you understand the playing cards better.
But what do we mean by probability?
A probability is a set of outcomes or a probable event. For example, while throwing a dice, the numbers that can come as a result are events.
The probability of picking out one card from a deck of cards is not all chance after all. This can be calculated. We have already learned this in school. But how many of you remember?
The probability of an event is the fraction of times the event occurs when the process is repeated several times. This fits in the case of selecting cards or throwing dice, and even booking a seat with many candidates applying at the same time.
For example, if we throw a dice, the chances of getting an even number is favourable outcome / total number of outcomes. Here favourable outcomes can be 1, 2, 3, that is it can occur three times whereas the total outcomes that can happen are 1,2,3,4,5,6 (six different outcomes). Now, the probability of getting a favourable outcome is 3/6 which can be further divided and become 1/2 or half.
In a nutshell, the probability is expressed by the ratio of the number of favourable outcomes/number of possible outcomes.
Theories of Probability
Conditional probability: It refers to the chances where several outcomes occur when another event has also occurred. It is known as the probability of B given A. It is written as P(B|A), where the probability of B depends on that of A.
For example, probability of pulling out a king of hearts from a deck of cards is –
favourable outcome (1) /number of possibilities (52)
If I change the deck of 52 cards and add another king of hears in it, my chances will increase as – a favourable outcome will become 2 and possibilities will remain 52.
Multiplication theorem of probability: This is the probability of an event happening that can be found using the definition of probability. A multiplication theorem can solve cases with both events.
This means, if A and B are two events such that P(A) ≠0 and P(B)≠0, then
P(A∩B) = P(A) * P(B|A) = P(B) *P(A|B)
Independent events: Two events such as A and B are called independent when there is no change in the happening of an event with the other one. This means, two events A and B are said to be independent if,
P(A|B) = P(A) where P(B)≠0.
P(B|A) = P(B) where P(A)≠0.
If P(A∩B) = P(A) * P(B), then it is an independent event
Bayes’ Theorem: It states the conditional probability of an event. It is based on the occurrence of another event and is equal to the chances of the second event given the first event is multiplied by its probability.
Bernoulli trials and binomial distribution: In the case of Bernoulli trials, there are only two possible outcomes but for the binomial distribution, the number of successes in a sequence of independent experiments.
How to Be a Master at cards aka Determine Probability
Let’s move on to how to master cards or how to understand the probability of cards and ace the game:
Step 1: Firstly, figure out the total number of cards you might pull. Write down all the possible cards. Mark the ones that you are likely to pull out
For example, if you are planning to take-
Hearts: 2, 3, 4, 5, 6, 7, 8, 9, 10, j, q, k, A
Clubs: 2, 3, 4, 5, 6, 7, 8, 9, 10, j, q, k, A
Spades: 2, 3, 4, 5, 6, 7, 8, 9, 10, j, q, k, A
Diamonds: 2, 3, 4, 5, 6, 7, 8, 9, 10, j, q, k, A
The total is 16 cards.
Step 2: Count the total number of cards in the deck. Since, we have one deck, so the total = 52
Step 3: Write the answer as a fraction. Divide it by 16 / 52. You get the probable score
Let’s Check Out Some Examples to Get a Better Understanding
Question: One card is drawn from a well-shuffled deck of 52 cards. If each outcome is equally likely, calculate the probability that the card will be:
(i) a diamond
(ii) not an ace
(iii) a black card (club or spade)
(iv) not a diamond
(v) not a black card
Solution:
When a card is drawn from a well-shuffled deck of 52 cards, the number of possible outcomes is 52.
(i) Let A be the event ‘the card drawn is a diamond’ Clearly the number of elements in set A is 13.
Therefore, P(A) = 13/52 = 1/4
Probability of a diamond card = 1/4
(ii) We assume that the event ‘card drawn is an ace’ is B. Therefore ‘card drawn is not an ace’ should be B′.
We know that P(B′) = 1 – P(B) = 1-4/52=1-1/13=12/13
(iii) Let C denote the event ‘card drawn is black card’
Therefore, number of elements in the set C = 26 which is P(C) = 26/52=1/2
Thus, probability of a black card = 1/2
(iv) We assumed in (i) above that A is the event ‘card drawn is a diamond’, so the event ‘card drawn is not a diamond’ may be denoted as A’ or ‘not A’
Now P(not A) = 1 – P(A) = 1-1/4=3/4
(v) The event ‘card drawn is not a black card’ may be denoted as C′ or ‘not C’.
We know that P(not C) = 1 – P(C) = 1-1/2=1/2
Therefore, probability of not a black card is 1/2
Question: Consider the experiment of drawing a card from a deck of 52 playing cards, in which the elementary events are assumed to be equally likely. If E and F denote the events ‘the card drawn is a spade’ and ‘the card drawn is an ace’ respectively, then,
Solution:
P(E) = 13/52 1/4 and P(F) 4/52 1/13
E and F is the event, the card drawn is the ace of spades, so that
P (E F) = 1/52
Hence, P (E|F) = P (E F)/P (F) 1/52 1/13 1/4
Since P(E) = 1/4=P (E|F), we can say that the occurrence of event F has not affected the probability of occurrence of the event E.
To learn about other topics taught in school, explained by News18, here is a list of other Classes With News18: Queries Related to Civics Chapter on ‘Elections’ | Sex Versus Gender | Natural Disasters | Wonderland of Letters | Civil Wars | Cryptocurrencies | Economy & Banks | Silk Route
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