State whether the events described are independent or dependent and determine the probability:


A card is pulled from a well shuffled deck of standard playing cards and is not put back in the deck. A second card is drawn from the deck. What is the probability that a spade was drawn both times?

Since the card is not replaced, the second draw is dependent on the first.

The probability of both/all events is found by multiplying the individual probabilities.

13/25 * 12/51 = ?

The events described are dependent because the second card drawn is affected by the outcome of the first card drawn.

To determine the probability of drawing a spade both times, we need to consider the number of favorable outcomes and the total number of possible outcomes.

First, let's determine the favorable outcomes: When the first card is drawn, there are 13 spades in a standard deck, so the probability of drawing a spade is 13/52.

Once the first card is drawn and not put back in the deck, there are now 51 cards remaining. Out of these, 12 are spades since one spade was already drawn. Therefore, the probability of drawing a spade on the second draw is 12/51.

To calculate the probability of both events occurring, we multiply the probability of the first event by the probability of the second event:

Probability of drawing a spade both times = (13/52) * (12/51) = 156/2652 = 13/221.

The probability of drawing a spade both times is 13/221.

To determine whether the events are independent or dependent, we need to consider if the outcome of the first event affects the outcome of the second event.

In this scenario, the events are dependent because the first card is not put back into the deck before drawing the second card. This means that the composition of the deck changes after the first card is drawn, potentially affecting the probability of drawing a spade on the second card.

To calculate the probability, we need to know the number of spades in a standard deck of playing cards. There are 13 spades in a deck, out of a total of 52 cards.

Let's break down the calculation step-by-step:

Step 1: Probability of drawing a spade on the first card.
There are 13 spades out of 52 cards in the deck, so the probability of drawing a spade on the first card is 13/52.

Step 2: Probability of drawing a spade on the second card, given that a spade was drawn on the first card.
After drawing the first card, there are now 51 cards left in the deck, and 12 spades remaining. So the probability of drawing a spade on the second card, given that a spade was drawn on the first card, is 12/51.

Step 3: Multiply the probabilities of the two events to find the probability of drawing a spade both times.
The probability of drawing a spade on the first card (13/52) multiplied by the probability of drawing a spade on the second card given that a spade was drawn on the first card (12/51) gives us the probability of drawing a spade both times: (13/52) * (12/51) = 1/17 ≈ 0.059 or 5.9%.

Therefore, the probability of drawing a spade both times is approximately 0.059 or 5.9%.