Playing Cards. One card is selected at random from an...

Playing Cards. One card is selected at random from an ordinary deck of 52 playing cards. Let $A=$ event a face card is selected, $B=$ event a king is selected, and $C=$ event a heart is selected. Find the following probabilities and express your results in words. Compute the conditional probabilities directly; do not use the conditional probability rule. a. $P(B)$ b. $P(B | A)$ c. $P(B | C)$ d. $P(B |(\text { not } A))$ e. $P(A)$ f. $P(A | B)$ g. $P(A | C)$ h. $P(A |(\text { not } B))$

Playing Cards. One card is selected at random from an ordinary deck of 52 playing cards. Let $A=$ event a face card is selected, $B=$ event a king is selected, and $C=$ event a heart is selected.
Find the following probabilities and express your results in words. Compute the conditional probabilities directly; do not use the conditional probability rule.
a. $P(B)$
b. $P(B | A)$
c. $P(B | C)$
d. $P(B |(\text { not } A))$
e. $P(A)$
f. $P(A | B)$
g. $P(A | C)$
h. $P(A |(\text { not } B))$

Key Concepts

Complementary Events

Complementary events are pairs of mutually exclusive events where the occurrence of one event means the other does not occur. Recognizing complementary events helps in calculating probabilities by considering the entire sample space.

Conditional Probability

Conditional probability measures the likelihood of an event occurring given that another event has already occurred. It is calculated directly by considering only the outcomes that satisfy the given condition.

Counting Techniques

Counting techniques involve methods such as enumeration or combinatorial analysis to determine the number of outcomes that satisfy certain conditions, which is crucial for calculating probabilities in finite sample spaces.

Events

An event is a collection of outcomes from the sample space that satisfy a specific condition. Events can be simple, such as drawing a specific card, or compound, such as drawing a card that meets multiple criteria.

Sample Space

The sample space is the complete set of all possible outcomes of an experiment. For example, knowing the sample space is essential when analyzing random draws from a deck of cards.

Probability

Probability quantifies the chance that an event will occur, defined as the ratio of the number of favorable outcomes to the total number of equally likely outcomes in the sample space.

Transcript

00:01 One card is selected at random from an ordinary deck of 52.

00:06 We are going to let a stand for the event.

00:14 A face card is selected.

00:25 We're going to let b stand for the event that a king is selected.

00:42 And we are going to let c be the event that a heart is selected.

00:52 And with this information, we are going to determine various conditional probabilities.

01:05 So let's start with part a.

01:11 Part a is the probability of event b.

01:17 So keep in mind, there are 52 cards, and probability is always going to be your favorable over your possible.

01:26 So in this case, there are 52 possible cards, and the favorable is that we select a king, and there are four kings in the deck.

01:39 So the probability of event b would be four out of 52, which simplifies to one -thirteenth, and as a decimal, that is approximately equal to 0 .077.

01:56 Part b, the probability of event b, given that we know a.

02:10 Okay, so we know that we are dealing with a face card.

02:16 So the face cards are the jacks, the queens, and the kings of each suit.

02:21 So we know there are 12 possible cards because there are 12 face cards.

02:31 Of those 12 face cards, four of them are kings.

02:36 We have the king of diamonds, the king of hearts, the king of spades, and the king of clubs, which simplifies down into one -third or approximately equal to a probability of 0 .33.

02:57 Let's go to part c.

03:00 What's the probability of b, given that we know.

03:06 C.

03:08 So this time we know the card is a heart.

03:12 Well, there happened to be 13 hearts in the deck, and we want a king as the favorable...

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