Simple Probability Problems?

Question

A. Ramon has exactly 26 cents in change in his pocket. What is the probability he has an odd number of coins in his pocket? Express your answer as a common fraction.


B. A football stadium has 7 gates for fans to enter or exit. The gates are numbered consecutively from 1 through 7. The probability of entering or exiting through each gate is equal. What is the probability Sylvia enters the stadium through an even numbered gate and leaves the stadium through an odd numbered gate? Express your answer as a common fraction.


C. ABCD represents Daniel’s 4-digit school locker number. The locker number is the square of a prime number. What is the probability the locker number is such that A < B < C < D? Express your answer as a common fraction.

Answer 1

A)There are only 12 ways to get 26 cents:

Q d n p total # of coins
1 0 0 1 ----- 2
0 1 0 16 ----- 17
0 1 1 11 ----- 13
0 1 2 6 ----- 9
0 1 3 1 ----- 5
0 2 0 6 ----- 8
0 2 1 1 ----- 4
0 0 1 21 ----- 22
0 0 2 16 ----- 18
0 0 3 11 ----- 14
0 0 4 6 ----- 10
0 0 5 1 ----- 6
4 out of the total 12 have an odd # of coins; that ratio reduced is 1/3
B) 3/7 *4/7 = 12/49
C)There are only 27 prime numbers that have a product with 4 digits when squared. Only three of which fits your criteria. (13, 17, 37)
Thus your probabilty would be 3/27 or 1/9
2*2 =0004 3*3 =0009 5*5 =0025 7*7 =0049
11*11=0121 13*13=0169 17*17=0289 19*19=0361
23*23=0529 29*29=0841 31*31=0961
37*37=1369 41*41=1681 43*43=1849 47*47=2209
53*53=2809 59*59=3481 61*61=3721 67*67=4489
71*71=5041 73*73=5329 79*79=6241 83*83=6889
87*87=7569 89*89=7921 91*91=8281 97*97=9409

Answer 2

Probability is defined as n(E) / n(S), where n(E) is the number of ways the event you're interested in can happen, and n(S) is the total number of events that can happen, that is, the number of events in the sample space. I'm not going to do your homework for you, because that would defeat the purpose of you learning this. But I'll give you a few clues as to how to do it yourself.

A. Write down all the ways you can get 26 cents in coin. That is, a quarter and a penny, two dimes a nickle and a penny, five nickles and a penny, etc. Count these up and that's n(S). Now note which of these ways involves an odd number of coins. That's n(E). Then your probability expressed as a common fraction would be n(E) / n(S).

B. With 7 gates numbered 1 thru 7, you have 4 gates with odd numbers and 3 gates with even numbers. So the probablity of an odd gate is n(E)/n(S) = 4/7 while the probability of an even gate is n(E)/n(S) = 3/7. Also you need the multiplication rule here, which states that when you have a sequence of independant events, the probabilty of the sequence is equal to the product of the probability of each event. So that would be 3/7 x 4/7.

C. Find n(S) by listing all the prime numbers whose squares is a four digit number. (There is a finite number of them, and they are all between sqrt 1000 which is a little shy of 32, and sqrt 10000 which is 100. To find them, use the sieve of Eristonese (sp?) or google it.) Then find n(E) by counting up the ones that have their digits in strictly ascending order. The probability again would be n(E) / n(S).

Answer 3

C:
step 1: four digits means the number if less than 100
step 2: there are 25 prime numbers between 1 and 100
step 3: for A 13 -> 0169
17 -> 0289
37 -> 1369
no other combinations fit.

Therefore, probability is 3/25.

Answer 4

A. You'd have to define what coins exist where Ramon has collected his change.
Let us say: 25, 10, 5, 1

Next, list all possible outcomes (number of coins making up 26 cents) and the number of ways that outcome can be achieved. For example, the outcome 26 can only be achieved one way: 26 coins worth 1 cent each.

If you allow 20 cent coins, then the distribution will be different.

Add the number of ways an odd outcome can be achieved and compare to all the ways 26 cents can exist.

If the probability of acquiring a coin is not affected by its value, then you're done. If this is a beginner's level statistics problem, this is what is normally used.

If the probability of getting a coin is affected by its value, then you have to model the probability for each value. For example, if Ramon is collecting coins from the sidewalk in front of a discount store, he is more likely to find pennies and nickels than he is of finding quarters.

Answer 5

B. 4/7 * 3/7 = 12/49

Answer 6

A: 4/13
B: 12/49
C: 3/25