probability......?

Question

A rack of clothes...5shirts and 4ties.
The 5 shirts -3 solid colors (blue, green, white)and 2 patterns(blue&green stripes, red&white stripes)
Match a shirt with a tie.... The 4 Ties- 3 solid colors (white, blue, yellow) and 1 pattern (green&blue stripe).
Can't wear 2 solids of the same color and if you wear a pattern, match it with a solid of one of the colors in the pattern.
WHAT IS THE PROBABILITY THAT YOU WILL PICK OUT A COMBINATION THAT INCLUDES BLUE???

Answer 1

Your best bet is to list out all the possible outfits you can make that fit your rules.

Pick a shirt first:
blue - this can be worn with a white, yellow, or green-blue stripe, but not blue tie [3 outfits here]

green - all ties can be worn with this [4 outfits]

white - only a blue or yellow tie can be worn [2 outfits]

blue-green stripes - blue tie [1 outfit]

red-white stripes - white tie [1 outfit]

So there are 11 total possible outfits, and 7 of them include the color blue somewhere in the outfit. Thus the probability is 7/11.

Answer 2

Well if you want to do this and show complete mathematical reasoning you need to find # of outcomes that satisfy your conditions, and # of total outcomes.

You would be satisfied with the following outcomes:

Blue Patterened Shirt & Any Solid Tie (BPS & ST)

Non-Blue Patterned Shirt & Blue Solid Tie (N-BPS & BST)

Blue Solid Shirt & Any Patterned Tie (BSS & PT)

Non-Blue Solid Shirt & Blue Patterned Tie (N-BSS & PT)

(*note that the probablity of Any patterened tie and a blue patterned tie are the same since there is only 1 patterened tie and it has blue in it)

Compute the porbablities of each scenario and add them up for your total probablity of an accepted outcome.

Start by computing probablity for each scenario based on information you are given :

First Scenario : ( BPS & ST)
(*note that & means multiply their probs together since they are represented by 1 event)

Chance of BPS = 1/5 ( because there is 1 blue patterned shirt and 5 shirts overall)
Chance of ST = 3/4 ( because there are 3 solid ties and 4 ties in total. *note that any solid tie would be acceptable in this scenario)

Now multiply both probablities to get the probablitiy of both occuring simulatneous (You have a Blue Patterned shirt and a Solid Tie)

1/5 * 3/4 = 3 /20 <--- probability of first scenario

For the other scenarios I will just do the math and not the explanations since I think you get it now :)

Second Scenario: (N-BPS & BST)
NBPS : 1/5
BST : 1/4
1/5*1/4 = 1/20 <-- probability of second scenario

Third Scenario:
BSS: 1/5
PT: 1/4
1/5*1/4 = 1/20

Fourth Scenario:
NBSS: 2/5
BPT: 1/4
2/5*1/4 = 2/20

NOW ADD UP ALL PROBABILITIES FOR THE SCENARIOS THAT YOU WANT :

3/20 + 1/20 + 1/20 + 2/20 = 5/20 = 1/4

Thus the probability that you get an outcome that matches your cinditions is 1/4 or 25% chance.

:)