What is the probability of getting a black card with an even number on it?

Probabilities are assigned to the outcomes of an experiment. We shall only consider experiments where all the outcomes are equally likely. Hence for drawing a card from a deck, each outcome has probability 1/52. The probability of an event is the sum of the probabilities of the outcomes in the event, hence the probability of drawing a spade is 13/52 = 1/4, and the probability of drawing a king is 4/52 = 1/13. Although the fact that the probability of getting a spade is 1/4 may make it seem that events are equally likely (since there are four suits), the probability of getting a face card is not equal to the probability of getting a number card. The only way to calculate the probability of an event is to sum the probabilities of the (equally likely) outcomes in the event. However, identifying what the equally likely outcomes are can be subtle as the following exercise illustrates.

Exercise: What are the equally likely outcomes for flipping a pair of coins? What are the equally likely outcomes for rolling a pair of dice?

Since the the probability of an event is the sum of the probabilities of the outcomes which comprise the event, one might assume that the probability of an event is the sum of the probabilities of any events which comprise that event. However, The probability of getting a black card or an ace [which we may denote as P(black or ace)] is not P(black) + P(ace) since the former is 28/52 (there are 26 black cards and 2 red aces) while the latter is 26/52 + 4/52. The discrepancy is due to the fact that the black aces are counted twice on the right hand side, once with the black cards and once with the aces. Correcting for the double counting provides the additive rule for arbitrary events: P(A or B) = P(A) + P(B) - P(A and B). Indeed 28/52 = 26/52 + 4/52 - 2/52 (there are two black aces).

Definition. Two events A and B are said to be mutually exclusive (or disjoint) if their intersection is empty (or equivalently, P(A and B) = 0). For example, the events getting a club and getting a one-eyed jack are mutually excllusive because the one-eyed jacks are the jacks of spades and hearts. The events getting a heart and getting a one-eyed jack are not mutually exclusive, because the jack of hearts is both a heart and a one eyed-jack. The additve rule for mutually exclusive events is P(A or B) = P(A) + P(B) (because P(a and B) which should be subtracted from the right hand side is equal to zero).

Definiton: A' (read A complement) is the set of outcomes which are not in A. It follows that P(A or A') = 1 and P(A and A') = 0.

Competencies:WHat is P(black or 2) What is P(Black and 2)? If you fip 3 coins, what is the probability of getting two heads? If you roll two dice, what is the probability that the sum of the pips will be 5?

A branch of mathematics that deals with the happening of a random event is termed probability. It is used in Maths to predict how likely events are to happen. The probability of any event can only be between 0 and 1 and it can also be written in the form of a percentage.

The probability of event A is generally written as P(A). Here, P represents the possibility and A represents the event. It states how likely an event is about to happen. The probability of an event can exist only between 0 and 1 where 0 indicates that event is not going to happen i.e. Impossibility and 1 indicates that it is going to happen for sure i.e. Certainty.

If not sure about the outcome of an event, take help of the probabilities of certain outcomes, how likely they occur. For a proper understanding of probability, take an example as tossing a coin, there will be two possible outcomes – heads or tails.

Formula of Probability

Probability of an event, P(A) = Favorable outcomes / Total number of outcomes

Some Terms of Probability Theory

There are different terms used in the probability that are not commonly used normally, terms like experiments, sample space, a favorable outcome, trial, random experiment, etc. Let’s take a look at their definitions in detail,

• Experiment: An operation or trial done to produce an outcome is called an experiment.
• Sample Space: An experiment together constitutes a sample space for all the possible outcomes. For example, the sample space of tossing a coin is head and tail.
• Favorable Outcome: An event that has produced the required result is called a favorable outcome. For example, If two dice are rolled at the same time then the possible or favorable outcomes of getting the sum of numbers on the two dice as 4 are (1, 3), (2, 2), and (3, 1).
• Trial: A trial means doing a random experiment.
• Random Experiment: A random experiment is an experiment that has a well-defined set of outcomes. For example, when a coin is tossed, a head or tail is obtained but the outcome is not sure that which one will appear.
• Event: An event is the outcome of a random experiment.
• Equally Likely Events: Equally likely events are rare events that have the same chances or probability of occurring. Here The outcome of one event is independent of the other. For instance, when a coin is tossed, there are equal chances of getting a head or a tail.
• Exhaustive Events: An exhaustive event is when the set of all outcomes of an experiment is equal to the sample space.
• Mutually Exclusive Events: Events that cannot happen simultaneously are called mutually exclusive events. For example, the climate can be either cold or hot. One cannot experience the same weather again and again.

The Possibility of only two outcomes which is an event will occur or not, like a person will eat or not eat the food, buying a bike or not buying a bike, etc. are examples of complementary events.

Some Probability Formulae

Addition rule: Union of two events, say A and B, then,

P(A or B) = P(A) + P(B) – P(A∩B)

P(A∪ B) = P(A) + P(B) – P(A∩B)

Complementary rule: If there are two possible events of an experiment so the probability of one event will be the Complement of another event. For example, if A and B are two possible events, then,

P(B) = 1 – P(A) or P(A’) = 1 – P(A).

P(A) + P(A′) = 1.

Conditional rule: When the probability of an event is given and the second is required for which first is given, then P(B, given A) = P(A and B), P(A, given B). It can be vice versa,

P(B∣A) = P(A∩B)/P(A)

Multiplication rule: Intersection of two other events i.e. events A and B need to occur simultaneously. Then 

P(A and B) = P(A)⋅P(B).

P(A∩B) = P(A)⋅P(B∣A)

What is the probability of drawing a black card from a well-shuffled deck of 52 cards?

Solution:

We know that a well-shuffled deck has 52 cards

Total number of black cards = 26

Total number of red cards = 26

Therefore probability of getting a black card= {total number of black cards in the deck}/{total number of cards in the deck}

= 26/52

= 1/2

So the probability of having black card is 1/2

Similar Questions

Question 1: What is the probability of getting a black queen or a diamond?

Solution:

Total number of cards=52

Number  of favorable cards that are black queen = 2

so, probability of getting a black queen= 2/52  

Total number of cards that are diamond=13

Therefore probability of getting a diamond= {13/52}    

Therefore, probability of getting a red ace or a spade, 

P(E) = probability of getting a  black queen+ probability of getting a diamond 

       = 2/52 +13/52

       = 15/32

Question 2: A bag has 20 balls of three colors, 8 balls of red color, 5 ball of blue color, and 7 balls of black color. If Ajay picks the ball randomly. What is the probability of Ajay picking up a red color ball from the bag?

What is the probability of drawing a even black card from a pack of 52 cards?

What is the probability of drawing a card with an even number?

How many even black cards are there?

How many even cards are black in a deck?