[Math] Probability statements in a true or false format

probabilitystatistics

The following questions are to be answered as preparation to my exam next Friday. I feel I understand the terms, such as "complement", "union", and "intersection", but when confronted with questions that necessitate this particular knowledge I struggle and mostly fail to answer them correctly. If anybody can give me some tips on how to go about answering these types of questions and the basics of this particular probability topic, I would appreciate it by incomprehensible amounts!

  1. Consider a fair six-sided die and the event of throwing a roll higher than 3. The complement of this event is {1,2,3}.
  2. Consider flipping a coin twice and the events A={TT, HH} and the event B={HT,TH}.
    The events A and B are mutually exclusive but not collectively exhaustive.
  3. After flipping a coin 100 times, you find that it has come Heads 61 times and Tails 39
    times. You therefore calculate that the probability of Heads is 61 percent. This is an
    application of the classical probability approach.
  4. When rolling two fair six-sided dice, the probability that the sum of the two rolls is 4, is higher than the probability that the sum is 5.

Best Answer

If you have an event, the complement can be thought of as NOT that event.

In case 1, the event is "scoring higher than 3 on a six sided dice". This means you can roll a 4, 5, or 6. This means the numbers that are NOT that event are 1, 2 and 3, thus the complement = {1, 2, 3}

Mutually exclusive means that if one event occurs, the other CANNOT occur, and vice versa. In case 2, if A occurs, then there is no way that B can, because that's how coins work!

Collectively exhaustive means that at least one of the events must occur. Obviously these two are not, if I got {HH, HH} then neither A or B has occurred.

In question 4, it may be useful to draw a table. On the top, write 1 2 3 4 5 6 for the first die, and vertically write 1 2 3 4 5 6 for the second die. If you add their scores where each row/column touches, then count how many are higher than 4, it will be greater than the amount that are higher than 5.

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