There are 25 red beads,x blue beads and y green beads in a box.When a bead is picked at random,the probability that it is blue is 1 over 4 and the probability that it is green is 1 over 3.
(a)Find the value of x and y.
(b)How many blue beads must be added so that the probability of picking a blue bead is 4 over 13?
Answer:a. X=15, y=20
b. 5
Thanks..
x=blue, y=green
x/(25+x+y)=1/4
Cross multiply and simplify
4x=25+x+y => 3x-y=25
y/(25+x+y)=1/3
Cross multiply and simplify
3y=25+x+y => -x+2y=25
Solving:
5x=75 => x=15
y=3*15-25=20
z+15/(60+z)=4/13
Cross multiply
13(z+15)=4(60+z)
9z=45
z=5
5 blue beads must be added.
To solve this problem, we can use the concept of probability.
(a) To find the values of x and y, we can set up the following equation:
Probability of picking a blue bead + Probability of picking a green bead + Probability of picking a red bead = 1
Given that the probability of picking a blue bead is 1/4 and the probability of picking a green bead is 1/3, we can substitute these values into the equation:
1/4 + 1/3 + Probability of picking a red bead = 1
To find the probability of picking a red bead, we subtract the sum of the probabilities for blue and green beads from 1:
1 - 1/4 - 1/3 = 12/12 - 3/12 - 4/12 = 5/12
We can now set up another equation based on the number of beads:
Number of blue beads + Number of green beads + Number of red beads = Total number of beads
Given that there are 25 red beads, we can substitute this value into the equation:
x + y + 25 = Total number of beads
We already know the probability of picking a blue bead is 1/4, so the number of blue beads can be expressed as:
x = 1/4 * Total number of beads
Similarly, since the probability of picking a green bead is 1/3, the number of green beads can be expressed as:
y = 1/3 * Total number of beads
Substituting the expressions for x and y into the second equation, we get:
1/4 * Total number of beads + 1/3 * Total number of beads + 25 = Total number of beads
Multiplying each term by the common denominator of 12 to eliminate the fractions, we obtain:
3 * Total number of beads + 4 * Total number of beads + 12 * 25 = 12 * Total number of beads
Simplifying the equation gives:
7 * Total number of beads + 300 = 12 * Total number of beads
Rearranging the equation, we have:
5 * Total number of beads = 300
Dividing both sides by 5 gives:
Total number of beads = 60
Now substituting this value back into the equations for x and y, we get:
x = 1/4 * 60 = 15
y = 1/3 * 60 = 20
Therefore, the values of x and y are x = 15 and y = 20.
(b) To find the number of blue beads that must be added so that the probability of picking a blue bead is 4/13, we can set up the following equation:
(Number of blue beads + Number of added blue beads) / (Total number of beads + Number of added blue beads) = 4/13
Substituting the values we know, we have:
(x + Number of added blue beads) / (60 + Number of added blue beads) = 4/13
Cross-multiplying the equation gives:
13 * (x + Number of added blue beads) = 4 * (60 + Number of added blue beads)
Simplifying the equation gives:
13x + 13 * Number of added blue beads = 240 + 4 * Number of added blue beads
Rearranging the equation, we have:
9 * Number of added blue beads = 240 - 13x
Dividing both sides by 9 gives:
Number of added blue beads = (240 - 13x) / 9
Substituting x = 15 (from part a), we get:
Number of added blue beads = (240 - 13 * 15) / 9 = (240 - 195) / 9 = 45/9 = 5
Therefore, to make the probability of picking a blue bead equal to 4/13, 5 blue beads must be added.