Assume that 2 cards are drawn in succession and without replacement from a standard deck of 52 cards. Find the probability that the following occurs: the second card is a 9, given that the first card was a 9.
Use the definition of conditional probability:
P(X9|9X) in the previous question.
P(X9∩9X)/P(9X)
=P(99)/P(9x)
=(4/52)*(3/51) / (4/52)
=3/51
Well, you know what they say, when life gives you lemons, ask for salt and tequila. But in this case, when life gives you a 9, you're interested in the probability of getting another 9.
Let's break it down. There are 52 cards in a deck, and if you've already pulled out a 9, you're left with 51 cards. Out of those 51, there are only 3 remaining 9s in the deck. So, the probability of drawing a 9 as the second card, given that the first card was a 9, is 3 out of 51.
But let's not stop there! We can simplify that fraction, and not just because we're feeling lazy. Both 3 and 51 can be divided by 3 (the other one doesn't really like division, but that's a different story). After reducing, the probability becomes 1 out of 17.
So, the probability of drawing a 9 as the second card, given that the first card was a 9, is a staggering 1 out of 17. Good luck chasing those nines, and remember, the real treasure was inside you all along.
To find the probability that the second card is a 9, given that the first card was a 9, we need to consider the number of favorable outcomes and the number of total outcomes.
Step 1: Determine the number of favorable outcomes.
When the first card is a 9, there are 3 remaining 9s in the deck (1 in each of the other 3 suits). So, there are 3 favorable outcomes.
Step 2: Determine the number of total outcomes.
After drawing the first card, there are 51 cards left in the deck. Therefore, there are 51 total outcomes for the second card.
Step 3: Calculate the probability.
Now, divide the number of favorable outcomes (3) by the number of total outcomes (51) to find the probability:
Probability = Number of favorable outcomes / Number of total outcomes
Probability = 3 / 51
Simplifying the fraction:
Probability = 1 / 17
So, the probability that the second card is a 9, given that the first card was a 9, is 1/17.
To find the probability that the second card is a 9, given that the first card was a 9, we first need to understand the total number of possible outcomes and the number of favorable outcomes.
Step 1: Determine the total number of possible outcomes.
When drawing two cards in succession without replacement from a standard deck of 52 cards, there are 52 options for the first card and 51 options for the second card. Hence, the total number of possible outcomes is 52 * 51.
Step 2: Determine the number of favorable outcomes.
Given that the first card was a 9, there are only 3 remaining 9s in the deck (since the 9 of hearts has already been drawn). Therefore, the number of favorable outcomes is 3.
Step 3: Calculate the probability.
The probability of an event occurring is given by the formula: favorable outcomes / total number of possible outcomes.
Plugging in the values we have determined, the probability that the second card is a 9, given that the first card was a 9, is:
Probability = 3 / (52 * 51)
= 3 / 2652
≈ 0.00113
Thus, the probability is approximately 0.00113 or 0.113%.