Hi, could you please help me? I have been stick on this question for a week and I have tried getting help from my teacher and parents, but I still cant get it. TIA
A standard deck of 52 cards consists of 4 suits (hearts, clubs, diamonds, and spades). If a student deals 3 cards from a standard deck, determine the probability to the nearest hundredth, that at least one card is a heart. Use the method stated below:
A.) Direct Reasoning (2 marks)
B.) Indirect Reasoning (2 marks)
52/4 = 13 hearts in the deck
52-13 = 39 are not hearts
probability that all 3 are H
= 13/52 *12/51*11/50 = .01294
probability that 2/3 are H
=13/52 * 12/51* 39/50 = .04588
{39 is number of non hearts}
probability that 1/3 is H
= 13/52 *39/51*38/50 = .14529
add
A) 0.2041
call h a heart
call x another suit
d = 52*51*50 = 132600
combos we could get
hhh =13*12*11/d = 1716/d
hhx =13*12*39/d = 6084/d
hxx =13*39*38/d = 19266/d
hxh =13*39*12/d = 6084/d
xxh =39*38*13/d = 19266/d
xhx =39*13*38/d = 19266/d
xhh =39*13*12/d = 6084/d
add to get
77766/d= 77766/132600 = .58647
That is the hard way
now
to get ALL x, no hearts
39/52*38/51*37/50 = 54834/d =.41353
1 - .41353 = .58647 !!! whew, the easy way
When I did it the easy way, I got a different answer, so went back and plodded through the hard way.
P(some hearts) = 1-P(no hearts)
= 1 - (39/52)(38/51)(37/50) = 0.58647
Certainly! I'll help you with this question step by step, explaining both direct reasoning and indirect reasoning methods.
Direct Reasoning:
To calculate the probability using direct reasoning, we need to find the total number of outcomes where at least one card is a heart and divide it by the total number of possible outcomes when dealing 3 cards.
Step 1: Find the total number of outcomes where at least one card is a heart.
There are two possible scenarios where at least one card is a heart:
1. One heart and two non-heart cards.
2. Two hearts and one non-heart card.
Scenario 1: One heart and two non-heart cards.
- Choose 1 heart card out of 13 hearts in the deck. (13C1)
- Choose 2 non-heart cards from the remaining 39 cards in the deck. (39C2)
Scenario 2: Two hearts and one non-heart card.
- Choose 2 heart cards out of 13 hearts in the deck. (13C2)
- Choose 1 non-heart card from the remaining 39 cards in the deck. (39C1)
Step 2: Find the total number of possible outcomes when dealing 3 cards.
Choose 3 cards out of 52 cards in the deck. (52C3)
Step 3: Calculate the probability.
The probability of at least one card being a heart is the sum of the outcomes from both scenarios divided by the total number of outcomes.
Probability = (Scenario 1 outcomes + Scenario 2 outcomes) / Total outcomes
Now, you can calculate the probability using these calculations.
Indirect Reasoning:
To calculate the probability using indirect reasoning, we will find the probability that no card is a heart and then subtract it from 1 to get the probability that at least one card is a heart.
Step 1: Find the probability that no card is a heart.
To calculate this, we need to find the number of outcomes where none of the cards are hearts and divide it by the total number of outcomes when dealing 3 cards.
To choose 3 non-heart cards, we only have 39 non-heart cards since we have removed the 13 hearts.
Step 2: Find the total number of possible outcomes when dealing 3 cards.
Choose 3 cards out of 52 cards in the deck. (52C3)
Step 3: Calculate the probability that no card is a heart.
The probability of no card being a heart is the number of outcomes where all the cards are non-hearts divided by the total number of outcomes.
Step 4: Calculate the probability that at least one card is a heart.
The probability of at least one card being a heart is obtained by subtracting the probability of no card being a heart from 1.
Now, using these calculations, you can find the probability using indirect reasoning.
I hope this helps you! Let me know if you need any further assistance with the calculations.