Question

Each prime number less than 12 is written on a card and all the cards are placed in a bag. If a card is drawn at random, its number recorded before being placed back in the bag, and a second card is drawn, what is the probability that the product of the two numbers is even?

Answer 1

"prime number" "the product of the two numbers is even"

The thing to realize here is that one of the numbers has to be two, because it is the only even prime. An odd times an odd is an odd.

Answer 2

So first lets realize that we have \(x \in \{2, 3, 5, 7, 11\}\)

Answer 3

@ArkGoLucky need more help?

Answer 4

yes

Answer 5

I thought 1 was prime

Answer 6

I think its that probability of drawing a two plus the probability of drawing a number other than 2 times the probability of drawing a 2 but that would be 1

Answer 7

and that doesn't make sense

Answer 8

1 isn't prime

Answer 9

It's the probability of drawing a 2 and another number in any order.

Answer 10

If 2 is drawn first, an even number is guaranteed

Answer 11

so 1/5

Answer 12

The total number of ways you can draw is: \( 5 \times 4\).
Because you start with \(5\) cards, and in the second drawing you only have \(4\).

Answer 13

@wio replacement of cards is done,

Answer 14

Ah, my bad...

Then it is \(5 \times 5\).

Answer 15

P(2) = 1/11
P(2,3,5,7,11) = 5/11

net probab = product of both..

Answer 16

this will be mutiplied by 2 since 2 can also be drawn in the second attempt..

Answer 17

The number of ways to draw a \(2\) first is: \(1 \times 5\)
The number of ways to draw a \(2\) second is: \(1 \times 5\)
The number of ways to draw a \(2\) first and second is: \(1 \times 1\)

We want to subtract the third one from the first two to avoid double counting.

Answer 18

ohh yes,, maybe i double counted a case,,please confirm..

Answer 19

So there are \[
1\times 5 + 1 \times 5 - 1 \times 1 = 5 + 5 - 1 = 9
\]

Answer 20

We get \(9\) ways to draw \(2\) at least once.
We have \(25\) total ways to draw.

So our probability is \(9/25\), unless I am mistaken.

Answer 21

why is it 1*5

Answer 22

It is \(1 \times 5\) because on our first draw, we have \(1\) way of drawing 2, on our second draw, we have \(5\) ways of drawing since anything we draw will be fine.
We multiply them because it's happening in series.

Answer 23

okay I kind of understand. Is there another way to solve this?

Answer 24

Usually there are many ways of solving things in probability.

Answer 25

For example, you can try to the number of ways to NOT draw 2, then go from there.

Answer 26

The number of ways to NOT draw 2 is: \(4\times 4 = 16\).
So the probability of NOT drawing 2 is \(16/25\).
We can then subtract this from 1 to find the ways to draw 2 ATLEAST ONCE.
\[
1 - 16/25 = 25/25 - 16/25 = 9/25
\]

Answer 27

okay that helps. thanks

Answer 28

You need to figure out how many ways you can pick at least one 2 vs total number of ways of picking the two cards.

How many different outcomes are there?
You can pick the first card in 5 ways, and you can pick the second card in 5 ways, so 5 x 5 = 25, and the total number of outcomes is 25.

How many ways can you pick at least one 2?
On first card:
If in your first card pick you pick a 2, there are 5 cards you can pick as second. This is 5 ways of picking a 2 as first card.
On second card:
Then if the first card is not a 2, there is only one way of picking a 2 in the second card. But since there are 4 ways you can have a card other than 2 as the first pick., there are 4 ways you can have no 2 as first card but a two as second card. This adds 4 outcomes with a 2 as second card.
5 outcomes (2 as first card) + 4 outcomes (2 as second card) = 9

Probability of even number = 9/25