Question

solve

Answer 1

In this case it is not too difficult to evaluate all of the options 'by hand'. To get exactly one pen and one pencil, the order doesn't matter, right ?

This means that {pen, pencil} and {pencil, pen} are both valid options. Let's start with the first one, the {pen, pencil} combination. As you have two items, you need to multiply to odds to find the overall probability. In this case:

P(pen) first : there are now 5 pens and 6 pencils in the bag, so the probability to grab a pen is 5/11, because any of the 5 pens will do. Once you grabbed this pen, the bag only contains 4 pens and 6 pencils. To grab a pencil from this lot is now 6/10. The overall probability is then P = 5/11 * 6/10.

I'll leave it up to you to calculate the probability for the option {pencil, pen}, which follows the same line of reasoning.

Your answer will be the sum of the two probabilities. Let me know if you have trouble getting this answer, OK ?

Answer 2

you wrote that answer will be the sum of the two probabilities.
but you have wrote
The overall probability is then P = 5/11 * 6/10.
i didnt get this point

Answer 3

Sorry, wasn't maybe quite clear on this. You have two ways to achieve the result, which are the {pen, pencil} and the {pencil,pen} option. I've shown how to calculate the probability for {pen, pencil}, for which you need to multiply two probabilities.

This comes down to the following:

Ptotal = P(pen, pencil) + P(pencil,pen).

In my initial response, I only showed you how to calculate P(pen, pencil), which requires multiplying two probabilities. If you do the same for P(pencil, pen), you only need to add those newly calculated probabilities to find Ptotal. Hope this clarifies without using too much symbology etc.

Answer 4

P (pen,pencil)= 5/11 * 6/10 = 30/110=3/11
P(pencil,pen) = ?

Answer 5

Let's go over the P(pencil, pen) together. If you start, you have 11 items in the bag, of which 5 are pens and 6 are pencils. If you grab an object from the bag, what is the probability that you have a pen ?

Answer 6

5/11

Answer 7

Super ! Now, once you have taken this pen, what is left in the bag ?

Answer 8

pencil

Answer 9

There is more than just a pencil left in the bag. You started with a bag of 5 pens and 6 pencils. If you remove one pen, you end up with a bag of 4 pens and 6 pencils, correct ?

Answer 10

yes

Answer 11

So if you would grab an item from that bag (the one from which you took the pen earlier), what are the probabilities for another pen and for another pencil ?

Answer 12

6/10

Answer 13

True, but is 6/10 the probability to grab a pen or a pencil then ?

Answer 14

5/11*6/10

Answer 15

Can you now calculate the probability that you first grabbed a pencil an secondly grabbed a pen ? It follows the same line of reasoning as above.

Answer 16

i think
total prob. = P (pen,pencil)+ P(pencil,pen)
== 5/11 * 6/10 +5/11 * 6/10
= 30/110+30/110
=3/11+3/11
=6/11

Answer 17

i have done right or wrong

Answer 18

choose 1 pen: \[\binom{5}{1}=5\]
choose 1 pencil: \[\binom{6}{1}=6\]
choose 2 items: \[\binom{11}{2}=55\]
probability: \[\frac{5\times 6}{55}=\frac{6}{11}\]
your solution is also correct, i am just showing you a different way to interpret the problem.

Answer 19

Sorry, back again. In this case you have made a slight mistake in the P(pencil, pen). It should be as follows:

Probability to start with a pencil is 6/11 because there are 6 pencils in the bag and you have a total of 11 items (sum of all pens and pencils) in the bag. If you've just taken a pencil from the bag, the bag now contains 5 pens and 5 pencils, so 10 items.

The probability of taking a pen this time is 5 pens / (5 pens + 5 pencils) = 5/10.

So if you want to calculate the probability that you first get a pencil and after that get a pen, you need to multiply 6/11 * 5/10. This also gives you 30/110. You correctly summed the probabilities, reaching an answer of 6/11.

Answer 20

@sirm3d, be careful, the value of P(pencil, pen) is the same as P(pen, pencil) but the calculation is not. Therefore msingh's calculation needs some tweaking.