You are shown five cards, Three have X written on them and two have Y. The cards are turned over so the X abd Y are no longer visible, and shuffled/ One card is picked and laid face up on the table. It is an X.
If the remaining cards are shuffled and another card is now laid on the table:
a) What do you expect this card to be?
b) Find the probability that this card is X
1. If in the eg. above, the card is turned up is a Y, and if the remaining cards are shuffled and another card is now laid on the table:
a) What do you expect this card to be?
b) Fins the probability that the card is X.
2. Two dice are rolled at the same time. Find the probabilty that
a) the 2nd die shows a 5, given that the first die showed a 5.
b) The sum is 6, given that the first die showed a 5.
c) the first die showed a 5, given that the sum is 6.
3A) A two digit no. is written down. Find the probability that
a) it is palindromic, given that the unit digit is a 5.
b) It is palindormic, given that the tens digit is a 5.
c) The unit digit is a 5, given that the tens digit is a 3.
d) It is divisible by 9, given that the tens digit is a 3.
e) The tens digit is a 3, given that it is divisible by 9.
4. Malcolm buys 4 tickets in a raffle in which 50 tickets are sold. Tickets are drawn for 1st and 2nd prizes.
a) What is the probability of Malcolm wining the first prize?
b) Work out what is the probability of him wining the second prize, given that he wins the first.
c) Find the probability of winning the second prize, given that Malcolm does not win 1st prize.
5. A card is selected from a 52 card pack. Find the probability that the card is :
a) a Jack, given that the card is a picture card,
b) a Heart, given that the card selected is red.
We do not do your homework for you. However, I will give you a start.
1ab) Among the remaining 4 cards, there are 2 X's and 2 Y's.
I hope this helps.
To find the answers to these probability questions, we need to apply some basic principles of probability. Let's go through each question step by step:
1. The first scenario is as follows:
a) Since one X card has already been revealed, there are now 2 X cards and 2 Y cards left. Therefore, we can expect the next card to have an equal chance of being X or Y.
b) To find the probability that the next card is X, we need to calculate the ratio of X cards to the total number of remaining cards. In this case, there are 2 X cards and 4 total remaining cards (2 X cards + 2 Y cards). So, the probability of drawing an X card is 2/4 or 1/2.
The second scenario starts with a Y card being turned up:
a) Similarly, since one Y card has been revealed, there are now 3 X cards and 1 Y card left. Therefore, we can expect the next card to be X.
b) The probability of drawing an X card remains the same as in the first scenario: 2/4 or 1/2.
2. For the dice rolling scenario:
a) If the first die shows a 5, it does not affect the outcome of the second die. Therefore, the probability of the second die showing a 5 is 1/6, which is the same as any other roll of the die.
b) To find the probability of the sum being 6, given that the first die showed a 5, we need to consider the possible outcomes. The first die can show a 5 with a probability of 1/6. Out of all the possible outcomes when the first die shows a 5, there is only one outcome (2nd die showing a 1) that gives a sum of 6. So, the probability is 1/6.
c) The probability that the first die shows a 5, given that the sum is 6, can be determined by looking at all the possible outcomes where the sum is 6. There are five such outcomes: (1, 5), (5, 1), (2, 4), (4, 2), and (3, 3). Out of these five outcomes, two have a 5 on the first die. Therefore, the probability is 2/5.
3. For the two-digit number scenario:
a) If the unit digit is a 5, there are 10 possible numbers (from 50 to 59) and only 5 palindromic numbers (55, 56, 57, 58, and 59). Therefore, the probability of the number being palindromic, given that the unit digit is a 5, is 5/10 or 1/2.
b) Similarly, if the tens digit is a 5, there are 10 possible numbers (from 50 to 59) and only 5 palindromic numbers (55, 56, 57, 58, and 59). Therefore, the probability is 5/10 or 1/2.
c) If the unit digit is a 5 and the tens digit is a 3, there are only 5 possible numbers (from 35 to 39) and only 1 palindromic number, which is 35. Therefore, the probability is 1/5.
d) To determine the probability that the number is divisible by 9, given that the tens digit is a 3, we need to count the number of two-digit numbers that are divisible by 9 and have a tens digit of 3. There are only two numbers that satisfy this condition: 36 and 45. Therefore, the probability is 2/5.
e) If the number is divisible by 9, there is only one possibility, which is 36. Out of all the numbers that are divisible by 9, only one has a tens digit of 3. Therefore, the probability is 1/1 or simply 1.
4. For the raffle scenario:
a) There are 50 tickets sold and Malcolm has 4 tickets. Hence, the probability of winning the first prize is 4/50 or 2/25.
b) Given that Malcolm has already won the first prize, the number of tickets remaining for the second prize is now 49 (since one winning ticket has been removed). Malcolm still has 4 tickets out of these 49, so the probability of winning the second prize is 4/49.
c) If Malcolm does not win the first prize, there are 49 tickets remaining. Since Malcolm still has 4 tickets, the probability of winning the second prize is again 4/49.
5. For the card selection scenario:
a) If the card is a picture card (face card), there are 12 picture cards in the deck (4 Jacks, 4 Queens, and 4 Kings). Therefore, the probability that the card is a Jack, given that it is a picture card, is 4/12 or 1/3.
b) If the card selected is red, there are 26 red cards in the deck (13 Hearts and 13 Diamonds). Since there are 52 cards in total, the probability that the card is a Heart, given that it is red, is 13/26 or 1/2.
Remember, probabilities are based on assumptions and can vary depending on the context and any additional information provided.