Probability?

Question

A box contains 5 white tags numbered 1 to 5, 5 red tags numbered from 1 to 5, and 5 green tags numbered from 1 to 5. If two tags are selected at random and without replacement what is the probability that:

a) both are the same number?
b) both are the same color?
c) they are consecutive numbers but of different colors?
d) they are consecutive numbers of the same color?


My initial guess for completing a & b is to take 5/15 x 4/14 + 5/15 x 4/14 + 5/15 x 4/14 = 0.2856, but I don't think this is completely correct, and I don't know what to do to differentiate between being the same color or the same number.

If you could please explain in more detail, I'd really appreciate it!

Answer 1

You can simplify the first cases:
a) Here you can just imagine that you pick a certain number (let's say '1'). What's the chance that the second tag matches? There are 2 other tiles with '1' on them (out of 14), so the probability is 2/14 or 1/7. Notice how the number of the tag doesn't matter.

Same thing for b)
Imagine you pulled a green tag. The chance that the second tag is also green (4 remaining) is 4/14 or 2/7. Again notice how the color of the first tag really doesn't matter.

c) Break down the 5 cases:
Draw a '1' --> second tile must be '2' of another color (2/14 = 1/7)
Draw a '2' --> second tile must be '1' or '3' of another color (4/14 = 2/7)
Draw a '3' --> second tile must be '2' or '4' of another color (again 2/7)
Draw a '4' --> second tile must be '3' or '5' of another color (again 2/7)
Draw a '5' --> second tile must be '4' of another color (1/7)

So the odds are (1/5)*(1/7) + (1/5)*(2/7) + (1/5)*(2/7) + (1/5)*(2/7) + (1/5)*(1/7). This is simplified to (1/5)(1/7 + 2/7 + 2/7 + 2/7 + 1/7)

= 1/5 * 8/7 = 8/35

d) is similar to c) but with the same color.
If 1 or 5, then only 1 other tile (out of 14) will match.
If 2 though 4, then 2 other tiles (out of 14) will match

1/5 * (1/14 + 2/14 + 2/14 + 2/14 + 1/14)
1/5 * (8/14)
1/5 * 4/7
4/35

Answer 2

A) the answer is arrived at by figuring how the numbered tags change with each draw. At first there are 3 of each number, so the odds of getting any one number is 5/15, or 1/3. Once the that number is drawn, there are only 2 of that number left, so that would mean there are 2/14 odds of drawing the same number, which is 1/7. The total odds would be 1/3*1/7 or 1/21.
b) using the same arguement as above for colors, the odds would be the same.
c). the odds here are a bit harder to compute, since there are two cases to consider. If one considers that the next "consecutive" number to the 5 is 1 then it is pretty straightforward. If not then there are a different set of calculations to do.
Case 1: 1 is consecutive to 5: drawing any number is 1/3. therefore there are only 4 possible tags to complete the sequence, so it would be 4/14 to pick from or 2/7. the answer would be 1/3*2/7.
Case 2: 1 is not consecutive to 5. If you draw a 1 or 5 then the chances are there are only 2 tags that work, thus 2/14 or 1/7 chances. answer is 1/3* 1/7
d) 1/3 * 2/14

Answer 3

a) both are the same number?
well the first time you can pick anything, the second one needs to match the first number, so if there are 3 of each number; then the answer is 2/14 = 1/7

b) both are the same color?
same logic to this one, first time you can pick anything the second one needs to match the first color. There are 5 of each color, so the answer is 4/14 = 2/7

c) they are consecutive numbers but of different colors?
I'm thinking you can pick 5, 4, 3 or 2 first because if you pick 1, there is no number to pick after it. So the first fraction is 12/15 = 4/5. We want a different color, so we only have 2 options for the second pick; 2/14 = 1/7
Multiply: 4/5 x 1/7 = 4/35


d) they are consecutive numbers of the same color?
4/5 is still the first fraction. Same color means 1/14 for the second fraction. Multiply 4/5 x 1/14 = 4/70 = 2/35


I hope my thinking is right :-)

Answer 4

a). after drawing the first tag, there are 14 tags left, 2 of which are the right number. prob = 2/14 = 1/7

b) after drawing the first tag, there are 14 tags left, 4 of which are the right color. prob = 4/14 = 2/7

c) if the first tag is numbered 1 or 5, only 2 of the remaining 14 tags work
if the first tag is 2,3, or 4, 4 of the remaining tags work (2 possible numbers, 2 colors).
2/5 * 1/7 + 3/5 *2/7 = 8/35

d) if the first tag is numbered 1 or 5, only 1 of the remaining tags works.
if the first tag is numbered 2, 3, or 4, then 2 of the remaining tags work
2/5 * 1/14 + 3/5 * 2/14 = 4/35

Answer 5

a - 5/15
b- 5/15
c- 3/15
d-1/15

w1 w2 w3 w4 w5
r1 r2 r3 r4 r5
g1 g2 g3 g4 g5

w1r1g1 w2r2g2 w3r3g3 w4r4g4 w5r5g5

w1r2g3 r1g2w3 g1w2r3 w2r3g4 r2g3w4 g2w3r4



n so on