Probability

Probability of drawing a seven of spades, from a 52 card deck, is 1/52.

It the the probability that the random variable in question takes any value up to and including the argument.

Suppose you have a random variable X and f(x) is the probability that X = x [that is, the rv X takes the value x].

If F(x) denotes the cumulative distribution function of X, then

F(x) is the sum of all f(y) where y <= x.

Thus, for a fair die,

F(1) = f(1) = 1/6

F(2) = f(1) + f(2) = 2/6

F(3) = f(1) + f(2) + f(3) = 3/6 and so on.

Note that

F(X) = 0 for X < 1,

F(a+b) where a is an integer in the interval [1,6] and 0<b<1 is F(a). Thus, for example, F(3.5) = F(3).

and F(x) = 1 for x >=6.

In the case of continuous probability distributions, the summation is replaced by integration.

For any particular trial, the total probability is 1.

Assuming a 52 card deck, 4 jacks / 52 cards or probability of 4/52 or 1/13 or 0.077.

1-digit number: 5 possibilities

2-digits number: 5 x 5 = 25 possibilities

So there are 30 numbers.

There are 12 face cards in a standard 52-card deck, a jack, a queen and a king of each of the four suits.

The probability of rolling a 2 is 1 in 6. The probability of rolling an even number is 3 in 6. The probability of doing both, on two rolls, is 3 in 36, or 1 in 12.

Fair dice have a 1/6 probability of showing any one of its numbers on top. This, is a simple example which most people are familiar with. It is therefore a useful way to introduce various concepts related to probability.

25

When there is at least one outcome in which both events occur simultaneously.

Event A: The next person to enter the mall is male

Event B: The next person to enter the mall is married

If a married man walks in next, then both events occur.

It means you have 100% chance or certain.

Assuming it is a normal fair cube, it is 1/6.

We have that stupid packet for science as well and we had the same question

probablity in what? a dice? if its a dice, it is 2/6

A perfect example of how theories are tested ! Theory predicts the outcome

of the experiment, and the purpose of the experiment is to test the theory.

If the outcome of the experiment is appreciably different from the one predicted

by the theory, (and if the experiment itself was valid), then the prediction is flawed,

and the theory must be either tweaked or trashed.

3/7

2, there could only be two outcomes. Heads or tails

As in any experiment you need to define a valid toss. So if you are examining the number of heads and tails you would limit the outcomes to these two states. However, in a real world experiment there may be a condition where the coin landed on a uneven surface and was tilted in some way. The number of tilted ways is infinite. You could also have the coin land on its edge which is unlikely but possible.

If there is a different number on each face of the die and no numbers a repeated it is a 1 in 10 chance.

There are a number of statistical tests that are designed for this purpose. The Chi-squared and Kolmogorov-Smirnov tests are two of the better known ways.

If you use 'G' to mean the number given away,

then the number remaining is (48 - G).

Principle of segregation

It can have multiple peaks, but most symmetrical distributions do not.

There are (64 x 63 x 62)/(3 x 2) possible sets of 3 squares;

there are 2 x (32 x 31 x 30)/(3 x 2) sets of 3 the same colour.

Of the 41664 possible threesomes, 9920 are identical in colour, a probability of 0.238

Added: This is identical to the probability of picking three odd or three even numbers from a bag containing the first 64 bingo balls!

Second add-on: I have amended the original answer which was wrong!

The alternative calculation is that after picking one square there are 31 of the remaining 63 which will be of the same colour and if that is so there will be 30 out of 62 to complete the trio. The calculation (simpler!) is thus 31/63 x 30/62 or 930/3906, a probability of 0.238.

There are four red cards that are a multiple of 5. The five of hearts, the five of diamonds, the ten of hearts, and the ten of diamonds.

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