Probability homework

Question

Two cards are drawn with out replacement from standard deck of 52 card. What is the probability that both cards are spades and are these events independent.

A.25/52 : Yes, they are independent

B. 3/52 ; No, they are dependent events

C. 1/17 ; No, they are dependent events

D. 1/16 ; Yes, they are independent events

Answer 1

There are 13 spades out of a deck of 52. The probability of the first event is 13/52. If you don't replace the probability of the second event is 12/51. You multiply the two events together and simplify the fraction which is 1/17. The events are dependent because the first event does influence the outcome of the second event since you don't replace the card.

Answer 2

Your question states that this is a homework problem, so I'm not going to solve it directly. Instead, I'll explain the two concepts present in the question and that should help you solve it on your own!

Events

Let's first discuss independent and dependent events. Event A is independent from event B when the occurrence of event A does not change the probability of event B occurring. Event B is dependent on event A when the occurrence of event A changes the probability of event B occurring. Let's look at some examples below:

Examples

1) Tossing a coin and rolling a die are independent events, because the result of the coin toss has no effect on the result of the die roll.

2) Drawing two cards from a deck with replacement are independent events, since the original card is returned to the deck after the first event. The probability of drawing the Ace of Hearts the second time is not dependent on which card is drawn the first time.

3) Picking different colored marbles from a bag without replacement are dependent events, since the total number of marbles changes from one event to the next. Since the total number of marbles changes each time one is picked, the probability of choosing specific colored marbles is different for each event.

Probabilities

Now let's discuss calculating the probabilities of more than one event occurring. It might be useful to consider conditional probability here, however, since the class is listed as algebra 2 I will not go deeper into that topic.

The probability of two events occurring can be found by multiplying the probabilities of each event occurring: P(1)*P(2), whether the events are independent or dependent. When using this method for dependent events, make sure to double check the fraction for P(2), since the second event is conditional on the first event in this case. See below for an example:

Example

Imagine sequentially drawing two cards from a standard deck of 52 cards without replacement. We want to determine the probability that both of these cards are aces.

The probability of drawing an ace from this deck is 4/52, since there are four aces and 52 total cards. 4/52 is our P(1) value. In order to determine the probability that we draw an ace the second time, without replacement, we must assume that an ace was drawn in the first event. Calculating the probability of drawing an ace the second time is now 3/51 - there are 51 total cards remaining and there are now 3 ace cards left in the deck (remember the ace from the first draw was not replaced). So 3/51 is our P(2) value.

So the probability of drawing two aces sequentially from a standard deck of 52 cards, without replacement, would be:

(4/52)*(3/51) = (12 / 2,652) = 1 / 221 .

Hope this helps!

Answer 3

Hi Leah.

The probability of the first card being a spade is 13/52 or 1/4.

The deck now has 12 spades of 51 remaining.

The probability of drawing a spade on the second draw only is 12/51.

The compound probability of drawing a spade on both the first draw and the second draw is

1/4*12/51= 12/204 which simplifies to 3/52. They are DEPENDENT because the first draw impacted the probability of the second draw.

Your answer is B.

Hope this helps.