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SIXTH GRADE / PROBABILITY / TEST 5.1.1

MATHEMATICS IS EASY. PROVIDED THAT ONE WORKS ENOUGH.
ENOUGH IS SKILL DEPENDENT, HAVE TO FIND "ENOUGH" FOR YOUR SIZE.

SOME OF OUR QUESTIONS ARE REALLY HARD, FOR THOSE THAT ARE SKILLED IN MATH.
SKIP THEM IN CASE NOT FOR YOU.

IF YOU DESIRE SO, WE CAN DISCUSS HARD/INTERESTING QUESTIONS IN OUR FORUM.

 PROBABILITY
    ♦ 
Sometimes events depends on very many parameters and some of them are unknown. (Theory of) Probability is our partly sufficient but necessary response to such situations.
    ♦ If we throw a coin in the air we can not control the forces on the coin and do not know if the outcome will be head or tail. But we know if we throw 1000 times 500 of each is to be expected.
    ♦ Experiment, outcome, sample space, event, independent event, discrete/ continuous space are important concepts to be known.
    Ref.-1
    Ref.-2
1. Muhammet Daghan German has dropped a stone and it hit the plate (form at left enlarged) below at the center and is scattered threafter. What is the probabilty of flight over section A that is congruent to the other three ?

!


a) 1
b) 1/3
c) 1/2
d) 1/4
4 congruent sectors should all have equal probabilities .
2. Same cast as in the previous question but his time the stone is given a kick by an experienced soccer player. Which of the choices shall be true about the probability of flight over the sector A after getting scattered ?

!


a) the point of hit has no discrete sample space.
b) 1/4
c) 1
d) 1/2
2The symmetry is now broken .
3. What will be the sample space, all possible outcomes, of an experiment of two students getting out of a coed classroom one after another if attention is paid to the order ?

a) {GG, GB, BG, BB}
b) {GG, BG, BB}
c) {GG, GB, BG, GB}
d) {GG, GB, BG, BB, GB}
The first may be a boy or a girl. In case of a girl there is two possibilities : GG and GB ...
This is an ill defined sample space though. We in fact, in the general case, have a need to the number of girls and boys in the class and determine our sample space based on ...
4. The same cast as in the previous question but the order of getting out is not considered or getting out is simultaneous, i.e. BG and GB are considered to be identical. Which of the choices will be correct about the number of elements of the sample space ?

a) 1
b) 3
c) 2
d) 4
This time the possible outputs are GG, BB and GB which is the same as BG .
Another ill defined sample space concentrating on one-shot outcome, human beings are identifiable not identical . Defining the sample space from this consideration gives us in real situation wrong probabilities even in case of 4 students in the class, 2G and 2 B.
5. What is the hit probability of the blue region if a dart is thrown many times and equal areas have equal hit probabilities ?

!


a) 2/5
b) 3/5
c) 1/3
d) 1/4
The purple region is 1/4 ths of the outer ring and has the same area with the yellow region in the middle of ...
y
6. Which is the highest probable sum if two dices are thrown together ?

In case of need for tips get the cursor on "y" or for help click at "ry1/2" .
RY1 RY2



a) 8
b) 5
c) 7
d) 6


Probable sum is the total of dice as explained in "http://gwydir.demon.co.uk/jo/probability/calcdice.htm" . The sample space is shown in ry1 .
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7. How many possible events are expected to occur if a coin is flipped thrice ?

a) 16
b) 256
c) 8
d) 128
Sample space is {{H,H,H},{H,H,T},{H,T,H},{H,T,T},{T,H,H},{T,H,T},{T,T,H},{T,T,T}} which has 8 elements. Every subspace, subset of this is set considered as an event. We se here the term subspace because the term (sample) space is generally used. What we have is called mathematically a set. To be a space is a norm needed.
8. Why not only single-element subsets/subspaces but all subsets are considered as an event ?

a) to assess all possibilities.
b) by definition.
c) the unions and intersections of (single element) subsets are to be considered.
d) to find the unions of the subsets.
If we just consider single element subsets which normally has only a fraction of the total probability 1, we may hardly get the total probability of all events.
9. Suppose a coin is tossed thrice and an 8 elemnent sample space came to existence. the ∅ is of course a subset of this constructed sample space Which event should correspond to ∅ ?

a) nonoccuring event
b) empty event
c) impossible event
d) whole event
Impossible events do have 0 probability .
10. Which is the certain event if a coin is tossed thrice ?

a) sample space
b) every event
c) each event
d) nearly certain event
Sample space is a set and is also a subset of itself. Either {H,H,H}, or {H,H,T}, or {H,T,H}, or {H,T,T}, or {T,H,H}, or {T,H,T}, or {T,T,H} or {T,T,T} i.e. one of every one-element subsets of the whole space is certainly to happen .
11. What is the probability of at least a tail if a coin is flipped thrice ?

a) 8/9
b) 7/8
c) 3/4
d) 9/10
{{H,H,H},{H,H,T},{H,T,H},{H,T,T},{T,H,H},{T,H,T},{T,T,H},{T,T,T}} .
y
12. What will the ratio of events be if a dart is thrown to each figure shown thrice (if hit probabilities of different colors are considered) ?

!


In case of need for tips get the cursor on "y" or for help click at "ry1/2".

RY1 RY2



a) 214
b) 2504
c) 254
d) 240
The sample space of the figure above has 83 = 512 elements; whereas the one below has 8 . For further help have please a look at ry1 .
13. The regular dodecahadron (12 congruent faces) has numbers from 1 to 12 on it's faces. What is the probability of 1 or an even number if rolled once ?

!


a) 5/12
b) 2/3
c) 7/12
d) 1/2
The probability of 1 is 1/12, and of an even is 1/2 olmaktadır. Since the proabilities are disjoint the answer is the sum ... .
14. All pieces of this grid have equal lngth x. How many possible shortest ways are there from A to B ?

!


a) 70
b) 14
c) 28
d) 56


First try structures with lesser elements like 2x2 or 3x3.
This is a rather hard question. If not solved, have please a look at the this page page/question-18 ("http://www.test-dr.com/education/6th-Grade-Mathematics/probability-5.1.1.html" /question-18) and it's helps.
15. What is the probability of a number less than 5 or an odd if dodecahedron is rolled twice ?

a) 1/3
b) 2/5
c) 3/5
d) 2/3
The probability of less than 5 is 4/12=1/3 .
The probability of an odd is 6/12=1/2 .
These probabilities would be summed if 1 and 3 were not both less than 5 and odd. In this case so doing they will be considered twice. The correct answer is 4/12 + 6/12 - 2/12 = 8/12 = ... .
16. Which choice will emerge if the case in previous question is generally formulated ?

a) O(A∪B)=O(A) + O(B) + O(A∩B)
b) O(A∪B)=O(A) + O(B) - O(A∩B)
c) O(A∪B)=O(A) - O(B) - O(A∩B)
d) O(A∪B)=O(A) + O(B) + O(A∩B)
The probabilities of both events will be summed and the probability of intersection will be subtracted .
17. What will be the probability of (2 , ......) if a dice and a dodecahedron are rolled together ?

!


a) 1/66
b) 1/84
c) 1/102
d) 1/72
Multiply the probabilities of ... .
y
18. What is the relation between the shortest distance travel possibilities from the nodes "x" to the corner "" and the Pascal Triangle ?

!


In case of need for tips get the cursor on "y" or for help click at "ry1/2" .
RY1 RY2


a) None.
b) Placed into the NW-SE diagonal nodes in correct order, they give the number of possible ways .
c) It is a method of constucting Pascal Triangle.
d) Pascal Triangle may be drawn this way.
Have please a look at ry1 .
19. Cyclist coming down does not know which is the right way out of the forest. He is supposed to ask the two girls sitting across the tree only one question to learn the correct way. One of the girls is a liar, she always lies and the other says always the truth. Which questions below are feasible/possible to ask ?

!


a) Which ways brings me out of the forest ?
b) Which way keeps me in this forest ?
c) If I ask your friend about the correct way out of this forest, which way will she indicate ?
d) All.
The third is little hard to ask, but all are logical questions .
20. Which is the correct question to be asked in case of the cyclist to get the right way if we assume that the girls each are informed about each other's behavior ?

a) choice "c"
b) choice "a"
c) "a" and "b" both
d) all
Negation of negative gives positive ...
21. What is the probability of at least 2 heads if a coin is flipped 4 times ?

a) 1/2
b) 11/16
c) 5/16
d) 1
sample space:{{H,H,H,H},{H,H,H,T},{H,H,T,H},{H,H,T,T},{H,T,H,H},{H,T,H,T},{H,T,T,H},{H,T,T,T}, {T,H,H,H},{T,H,H,T},{T,H,T,H},{T,H,T,T},{T,T,H,H},{T,T,H,T},{T,T,T,H},{T,T,T,T}} .
y
22. What is the probability of attaining a square of a natural number if the algebraic expression 2·p4 - 7·p2 +1 is evaluated for primes less than 10 ?

In case of need for tips get the cursor on "y" or for help click at "ry1/2" .

RY1 RY2



a) 1/3
b) 2/3
c) 1/4
d) 1/2


23. 80 kg./acre of wheat has been thrown to a field of an area of 5 acres. How many kg. of wheat is to be expected to be eaten by the birds if probability of being eaten is 1/10 ths ?

a) 40
b) 30
c) 80
d) 20
Total thrown is 400 kg .
24. Supposing that we have a keyboard having the letters A to J on it's first (upper) row. How many times do we normally need (or are we supposed) to press on upper row letters of this board till we get the word "dede" ? ?

a) 1 000
b) 100
c) 10 000
d) 100 000
The probability of first "d" is 1/10 ths and we have 4 letters .
25. What is the probability of converting all 10 kicks to goals for a very good kicker in case of ten penalty kicks with no goalkeeper between the goalposts ?

a) 10
b) 1
c) 9/10
d) 8/10
Probability is always less than or equal to 1 .
This is the feedback!

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