MGMT2262 Tutorial Sheet 2 Solutions

To begin, we create a crosstab.

	Do own return	Not do own return	Total
Refund	0.68	0.08	0.76
No refund	0.19	0.05	0.24
Total	0.87	0.13	1.0

1) P(return or refund) = P(return) + P(refund) – P(return and refund) = 87% + 76% - 68% = 95%

2) P(refund | return) = P(return and refund)/P(return) = 68/87 = 78.16%

3) P(refund | not own return) = P(not own return and refund)/P(not own return) = 8/13 = 61.54% which is less than 78.16% from question 2. Those who don’t do their own return are not more likely to get a refund than those who do.

4) Left side: P(return and refund) = 68%

Right side: P(return)*P(refund) = (0.76)(0.87) = 66.12%

The two sides are not equal. Therefore, getting a refund depends on whether or not someone does their own return. The alternate solution is to note that P(refund | return) = 78.16% (from question 2) does not equal P(refund) = 76% and the same conclusion is drawn.

5) P(balance < $1000) = (806 + 2690)/10000 = 0.3496 = 34.96%

6) P(balance < $1000 | income between $35K and $50K) = (508 + 1377)/3029 = 0.6223 = 62.23%

7) P(balance < $1000 | income < $75K) = (107 + 508 + 191 + 169 + 1377 + 826)/(276 + 3029 + 3241) = = 0.4855 = 48.55%

8) P(income at least $50K | balance at least $1000) = (1419 + 911 + 254 + 805 + 1229 + 742)/(3474 + 3030) = 0.8241 = 82.41%

9) For each income group, we need to compute the percentage that has a balance of $1000 or more:

< 35K	35K – 50K	50K – 75K	75K – 100K	100K +
0/276 = 0	(890 + 254)/ 3029 = 0.3777	(1419 + 805)/ 3241 = 0.6862	(911 + 1229)/ 2394 = 0.8939	(254 + 742)/ 1060 = 0.9396

Thus the $100K+ group has the largest percentage of those whose balance is $1000 or more. As we can see from the calculations, the percentage who have a balance of at least $1000 increases with each income group.

10) No-one with an income of at least $75,000 has a credit card balance under $500. No-one with an income below $35,000 has a credit card balance of $1000 or more.

11) P(recycle) = (0.45)(0.12) + (0.3)(0.18) + (0.25)(0.3) = 0.183 = 18.3%. It should also be noted that P(don’t recycle) = 100% - 18.3% = 81.7%

12) P(recycle milk jugs) = (0.45)(0.12)(0.04) + (0.3)(0.18)(0.08) + (0.25)(0.3)(0.4) = 0.03648 = 3.648%.

13) P(not recycle milk jugs) = (0.45)(0.12)(0.96) + (0.3)(0.18)(0.92) + (0.25)(0.3)(0.6) = 0.14652 = 14.652%. It should be noted that P(recycle milk jugs) + P(not recycle milk jugs) = 18.3% which is the total percentage of those who recycle from question 11. Then P(recycle milk jugs | recycle) = 3.648/18.3 = 19.93%

14) P(income ³ 60K) = (0.246)(0.702) + (0.754)(0.354) = 43.9608% or roughly 43.96%.

15) From question 14, we can deduce P(income < 60K) = 100% - 43.9608% = 56.0392%. From the previous question, we also know that P(no university and income ³ 60K) = (0.754)(0.354) = 26.6916%. We can now fill in the crosstab:

	University	No university	Total
60K+	17.2692	26.6916	43.9608
<60K	7.3308	48.7084	56.0392
Total	24.6	75.4	100

Then P(no university | <60K) = 48.7084/56.0392 = 86.92%

16) 36P6 = 1,402,410,240. The alternate solution is 36x35x34x33x32x31.

17) One solution is 26x25x34x33x32x31 = 723,465,600, since we have 26 choices for the first character of the password, then 25 choices for the last character of the password and finally 34 choices for the remaining 4 characters of the password. The alternate solution is 26P2 x 34P4 = 650 x 1,113,024 = 723,465,600.