Respuesta :
Answer:
So, the probability that Gary draws 2 pawns and 1 knight (in any order) from the bag with white pieces is
1
20
20
1
.
Step-by-step explanation:
To find the probability of drawing 2 pawns and 1 knight (in any order) from the bag with white pieces, we need to calculate the total number of ways to draw 3 pieces and the number of favorable ways to draw 2 pawns and 1 knight.
Total number of ways to draw 3 pieces:
There are a total of 8 + 2 + 2 + 2 + 1 + 1 = 16 white pieces in the bag.
So, the total number of ways to draw 3 pieces is given by the combination formula:
Total ways
=
(
16
3
)
Total ways=(
3
16
)
Number of favorable ways to draw 2 pawns and 1 knight:
There are 8 pawns and 2 knights in the bag. We need to draw 2 pawns and 1 knight in any order. The number of ways to do this is the product of the combinations:
Favorable ways
=
(
8
2
)
×
(
2
1
)
Favorable ways=(
2
8
)×(
1
2
)
Probability:
The probability of an event is given by the ratio of favorable ways to total ways:
Probability
=
Favorable ways
Total ways
Probability=
Total ways
Favorable ways
Now, let's substitute the values into the formulas and calculate:
Total ways
=
(
16
3
)
=
16
!
3
!
(
16
−
3
)
!
=
16
×
15
×
14
3
×
2
×
1
=
560
Total ways=(
3
16
)=
3!(16−3)!
16!
=
3×2×1
16×15×14
=560
Favorable ways
=
(
8
2
)
×
(
2
1
)
=
8
!
2
!
(
8
−
2
)
!
×
2
!
1
!
(
2
−
1
)
!
=
28
×
2
2
=
28
Favorable ways=(
2
8
)×(
1
2
)=
2!(8−2)!
8!
×
1!(2−1)!
2!
=
2
28×2
=28
Probability
=
28
560
=
1
20
Probability=
560
28
=
20
1
So, the probability that Gary draws 2 pawns and 1 knight (in any order) from the bag with white pieces is
1
20
20
1
.
Alright, let's calculate the probability!
The total number of ways Gary can draw 3 pieces from the bag of white pieces is C(16, 3), which is equal to 16! / (3! * (16-3)!) = 560.
The number of favorable outcomes, which is the number of ways Gary can draw 2 pawns and 1 knight, is C(8, 2) * C(2, 1), which is equal to (8! / (2! * (8-2)!) * (2! / (1! * (2-1)!)) = 28.
Now, we can calculate the probability by dividing the number of favorable outcomes by the total number of possible outcomes:
Probability = Number of favorable outcomes / Total number of possible outcomes = 28 / 560 = 0.05 or 5%.
So, the probability that Gary draws 2 pawns and 1 knight (in any order) from the bag of white pieces is 5%.
The total number of ways Gary can draw 3 pieces from the bag of white pieces is C(16, 3), which is equal to 16! / (3! * (16-3)!) = 560.
The number of favorable outcomes, which is the number of ways Gary can draw 2 pawns and 1 knight, is C(8, 2) * C(2, 1), which is equal to (8! / (2! * (8-2)!) * (2! / (1! * (2-1)!)) = 28.
Now, we can calculate the probability by dividing the number of favorable outcomes by the total number of possible outcomes:
Probability = Number of favorable outcomes / Total number of possible outcomes = 28 / 560 = 0.05 or 5%.
So, the probability that Gary draws 2 pawns and 1 knight (in any order) from the bag of white pieces is 5%.
