Probability with discrete random variable example

Hugo plans to buy packs of baseball cards until he gets the card of his favorite player but he only has enough money to buy at most four packs suppose that each pack has probability 0.2 of containing the card Hugo is hoping for let the random variable X be the number of packs of cards Hugo buys here is the probability distribution for X so it looks like there is a 0.2 probability that he buys 1 pack and that makes sense because that first pack there is a 0.2 probability that it contains his favorite players card and if it does at that point he'll just stop he won't buy any more packs now what about the probability that he buys two packs well over here they give it a 0.16 and that makes sense there's a point 8 probability that he does not get the card he wants on the first one and then there's another point 2 that he gets it on the second one so 0.8 times 0.2 does indeed equal 0.16 but they're not asking us to calculate that they give it to us then the probability that he gets three packs is 0.12 eight and then they've left blank the probability that he gets four packs but this is the entire discrete probability distribution because Hugo has to stop at four if even if he doesn't get the card he wants at four on the fourth pack he's just going to stop over there so we could actually figure out this question mark by just realizing that these four probabilities have to add up to one but let's just first answer the question find the indicated probability what is the probability that X is greater than or equal to two what is the probability remember X is the number of packs of cards Hugo buys I encourage you to pause the video and try to figure it out so let's look at the scenario as we're talking about probability that our discrete random variable X is greater than or equal to two well that's these three scenarios right over here and so what is their combined find probability well you might want to say hey we to figure out what the probability of getting exactly four packs are but we have to remember that they all add up to 100% and so this right over here is zero point two and so this is zero point two the other three combined have to add up to zero point eight zero point eight plus zero point two is one or a hundred percent so just like that we know that this is zero point eight if for kicks we wanted to figure out this question mark right over here we could just say that look have to add up to one so we could say the probability of exactly four is going to be equal to one minus 0.2 minus 0.16 - 0.128 I get one minus point two minus point one six minus point one two eight is equal to zero point 5 1 2 is equal to 0.51 to zero point five one two you might immediately say wait wait this seems like a very high probability there's more than a 50% chance that he buys four packs and he has to remember he has to stop at four even if on the fourth he doesn't get the card he wants he still has to stop there so there's a high probability that that's where we end up there is a little less than 50% chance that he gets the card he's looking for before that point