Prepare a probability distribution for the experiment. Let x...

Prepare a probability distribution for the experiment. Let x represent the random variable, and let P represent the probability. Four cards are drawn from a deck. The number of red tens is counted. A. x | P(x) 0 | 0.145 1 | 0.145 2 | 0.851 B. x | P(x) 0 | 0.851 1 | 0.145 2 | 0.005 C. x | P(x) 0 | 0.719 1 | 0.280 2 | 0.001 D. x | P(x) 0 | 0.001 1 | 0.280 2 | 0.719

Transcript

00:01 All right, so we have an experiment, and we need to read the experiment carefully to really understand what's happening.

00:07 We have four cards are drawn with replacement, and the number of black cards are noted, and the probability distribution is given here, and we want to fill in these blanks.

00:16 So the key thing to note is the with replacement.

00:29 It's a key thing to note, because it's nice to understand what's happening in these experiments to really understand how to find the distribution.

00:35 So we have our 52 cards, and then what we're doing is we're drawing a card.

00:41 Looking at it whether it's red or black, noting it and then putting it back, drawing it again.

00:49 Now the 52, same probability of picking a card and noting if it is it put a black, put it back, do it again, a third time, red or black, put it back, fourth time, red the black, put it back.

01:03 And so then what we're doing is keeping track of how many black cards we saw.

01:08 We had either 0, 1, 2, 3, or 4.

01:11 We're going to find probabilities of that.

01:14 And so this is the makings of a binomial distribution because all the probabilities are the same.

01:22 The probability of seeing a red or black card is the same for each trial.

01:32 And it's safe to assume it's independent.

01:35 That drawing this one has no impact on the outcome of this one.

01:42 And the formula is the following p of x, is n choose x times p to the x times one minus p to the n minus x all right so uh the probability of a black card or red card is the same in a 52 card deck probability of black card is equal to 26 out of 52 which is one half and n is four and so we'll fill it into our formula here and we'll simplify it so i'll do it in magenta here.

02:24 So p of x equals 4 choose x times 1 1⁄2 to the x times 1 minus 1⁄2 to the 4 minus x.

02:39 We can simplify this.

02:41 1 minus 1⁄2 is a half.

02:43 So this can be simplified.

02:51 And then by the laws of x -ponds, we can combine these because they're the same base.

02:55 So x minus x, those are cancel out, leaving just the 4...

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