in the the first video introducing random variable. Please explain why probability of X=9 is 05^9*0.5 (where X =number of heads in 10 tosses),

the same as the probability of X=10

I do not understand why it is not 0.5^9 common sense it should be greater than P(X=10)

Sorry, I do not understand your notation.

Can you give the video title and a time mark?

video title random variables at minute 2:41

Thanks, I’ll review it and provide a reply shortly.

At 1:57, the instructor gives the most important clue:

Since the coin tosses are independent, each event has a probability of 0.5, and the probability of a series of independent events is the product of the individual probabilities.

For example:

So if you want X = 3 (all three tosses are Heads),

-that’s three coin tosses

- the probability of each toss being a head is 0.5
- and the probability of all three being a head is 0.5 * 0.5 * 0.5, or 0.5^3.

In general, the probability of X = N is 0.5 ^ N

For X = 3, the probability is 0.125

For X = 10, the probability is 0.000977.

Now for the case where there are 9 heads and 1 tails in 10 tosses:

Here is the image from the slide in that video (with my added markup):

Notice that 0.5^9 * 0.5 is exactly equal to 0.5^{10}. Since they have the same base, the exponents simply add together.

In this experiment, we do not care about the order of the outcomes, only the totals after 10 coin flips.

Since each flip is independent, the order does not matter. Notice that the 3rd row in the image has the same probability as the 2nd row, because they both consider X = 9.

Both heads and tails have the same probability: 0.5.

So, the probability of any specific sequence of Heads and Tails with X = 9 will be the same.

Now if the question is what are the chances of getting any of the sequences that have nine heads and one tail, there are 10 such possible outcomes.

The probably of getting one of those outcomes is 10 / 2^{10} = 0.009766, since there are 10 outcomes with one tail, and 2^{10} possible outcomes in total.

But that’s a different question than the instructor was discussing.