A standard deck of cards has had all the face cards (Jacks, queens, and kings) removed so that only the ace through ten of each suit remains. A game is played in which two cards are drawn (without replacement) from this deck and a six-sided die is rolled. For the purpose of this game, an ace is considered to have a value of 1.
I know that the possible outcomes for this game is 9360.
but I get this question wrong.
b) Find the probability of each of the following events
i)one even card is drawn and an even number rolled
Second question I need help is this
A bag contains four red, three green, and five yellow marbles. Three marbles are drawn, one at time, without replacement. Determine the probablity that the order in which they are selected is
a) yellow, red, green
b)yellow, green,green
2ii is correct.
For 1B, there are two cases.
He draws 1 even followed by 1 odd:
1 even: 20 out of 40 cards
1 odd : 20 out of 39 cards.
Probability : (20/40)*(20/39)=400/1560=10/39
Similarly, 1 odd followed by 1 even:
1 even: 20 out of 40 cards
1 odd : 20 out of 39 cards.
Probability : (20/40)*(20/39)=400/1560=10/39
Total of two cases : (10+10)/39 = 20/39
For the die, there are 3 even out of 6, so the probability is 1/2.
Overall probability = (20/39)*(1/2) = 10/39.
I still don't understand 1b.. the answer is 10/39 but i don't know...
For 2ii) its
Yellow: 5/12
Green:3/11
Green:2/10
(5/12)*(3/11)*(2/10= 1/44
Thank you! I understand now.
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To find the probability of each event, we need to first determine the total number of possible outcomes and then calculate the number of favorable outcomes for each event.
1) Finding the total number of outcomes:
Since two cards are drawn without replacement from a deck of 40 cards, the total number of possible outcomes can be calculated as follows:
Total outcomes = (Number of ways to draw the first card) * (Number of ways to draw the second card) * (Number of possible rolls of the die)
Total outcomes = (20/40) * (19/39) * 6
Total outcomes = 3
2) Finding the number of favorable outcomes for each event:
i) One even card is drawn and an even number rolled:
For this event, we need to consider the following possibilities:
- An even card drawn from the deck: There are 10 even cards in the deck.
- An even number rolled on the die: There are 3 even numbers (2, 4, 6) on a six-sided die.
Number of favorable outcomes = (Number of ways to draw an even card) * (Number of ways to roll an even number)
Number of favorable outcomes = (10/40) * (3/39) = 1/13
3) Calculating the probabilities:
To calculate the probability of an event, divide the number of favorable outcomes by the total number of outcomes.
i) Probability of one even card drawn and an even number rolled:
Probability = (Number of favorable outcomes) / (Total number of outcomes)
Probability = (1/13) / 3
Probability = 1/39
For the second question:
1) Finding the total number of outcomes:
Since three marbles are drawn without replacement, the total number of possible outcomes can be calculated as follows:
Total outcomes = (Number of ways to draw the first marble) * (Number of ways to draw the second marble) * (Number of ways to draw the third marble)
Total outcomes = (5/12) * (4/11) * (3/10)
Total outcomes = 1/22
2) Finding the number of favorable outcomes for each event:
a) Yellow, red, green:
Number of favorable outcomes = (Number of ways to draw a yellow marble) * (Number of ways to draw a red marble) * (Number of ways to draw a green marble)
Number of favorable outcomes = (5/12) * (4/11) * (3/10) = 1/22
b) Yellow, green, green:
Number of favorable outcomes = (Number of ways to draw a yellow marble) * (Number of ways to draw the first green marble) * (Number of ways to draw the second green marble)
Number of favorable outcomes = (5/12) * (3/11) * (2/10) = 1/44
3) Calculating the probabilities:
To calculate the probability of an event, divide the number of favorable outcomes by the total number of outcomes.
a) Probability of yellow, red, green:
Probability = (Number of favorable outcomes) / (Total number of outcomes)
Probability = (1/22) / (1/22) = 1/1 = 1
b) Probability of yellow, green, green:
Probability = (Number of favorable outcomes) / (Total number of outcomes)
Probability = (1/44) / (1/22)
Probability = 1/2
Therefore:
a) The probability that the order in which the marbles are selected is yellow, red, green is 1.
b) The probability that the order in which the marbles are selected is yellow, green, green is 1/2.
1a. "Two cards are drawn" means that the order is not important.
So the outcome for the cards is
C(40,2)
= 40*39/2!
= 780
Multiply by the 6 outcomes of the independent event of the die to give 4680.
b. I interpret "one even card is drawn" to mean "exactly one even card is drawn" and not "at least one even card is drawn".
Out of the 4 possible outcomes:
Odd/Odd, even/even, even/odd, and odd/even, the probability that exactly one even card is drawn is 2/4. Similarly, the probability for an even number on the die is 3/6.
So the joint probability can be obtained by the multiplication rule.
2a.
Given 4 red, 3 green and 5 yellow marbles (total = 12).
Three are drawn without replacement
a) To be drawn in order YRG:
To get a yellow, there are 5 out of 12
To get a red on the next, there are 4 out of 11
To get a green on the next, there are 3 out of 10.
The probability is therefore:
(5/12)*(4/11)*(3/10)
ii) left to you as an exercise.