Probability

Step 1: Understanding the situation.
The total number of cards is 10, numbered from 1 to 10. The probability is conditional on the fact that the drawn card is more than 3. Therefore, we are only concerned with the cards numbered 4 through 10.

Step 2: Finding the favorable outcomes.
The even numbers greater than 3 are 4, 6, 8, and 10. So, the favorable outcomes are 4, 6, 8, and 10, which are 4 even numbers.

Step 3: Finding the total possible outcomes.
The cards greater than 3 are numbered 4, 5, 6, 7, 8, 9, and 10, giving us a total of 7 possible outcomes.

Step 4: Calculating the probability.
The probability that the card drawn is even given that it is greater than 3 is the ratio of favorable outcomes to total outcomes:

Step 5: Conclusion.
Thus, the probability that the card drawn is even, given that it is greater than 3, is .

Step 1: Understanding the situation.
There are 15 balls in total, 10 black and 5 white. We need to find the probability that both balls drawn are black, considering that the draws are without replacement.

Step 2: Finding the probability of drawing the first black ball.
The probability of drawing a black ball on the first draw is:

Step 3: Finding the probability of drawing the second black ball.
After drawing the first black ball, there are now 9 black balls left and 14 balls in total. The probability of drawing a black ball on the second draw is:

Step 4: Calculating the overall probability.
Since the draws are without replacement, the total probability is the product of the individual probabilities:

Step 5: Conclusion.
Thus, the probability that both drawn balls are black is .

Step 1: Understanding independent events.
For two independent events, the conditional probability is equal to the probability of , i.e.,

Step 2: Applying the given values.
Since , we have:

Step 3: Conclusion.
Thus, .