## Task 6B - 191246B

In a dark room, there are 20 individual gloves of the same size, namely

- 5 white gloves for the right hand
- 5 white gloves for the left hand
- 5 black gloves for the right hand
- 5 black gloves for the left hand

Two gloves are considered a matching pair if and only if they are of the same color and one is for the right hand, the other for the left hand.

A draw is understood to be the removal of a single glove, without the possibility of selecting by color and form. A game of $n$ draws consists of performing $n$ draws in succession, collecting the gloves thus removed, and only after these $n$ draws determining whether there is (at least) one matching pair among the $n$ removed gloves. The game is considered successful if and only if this is the case.

a) Determine the smallest natural number $n$ with the property that a game of $n$ draws is guaranteed to be successful!

b) Determine the smallest natural number $k$ with the property that a game of $k$ draws is successful with a probability greater than 0.99!
According to the given information, we have four groups of 5 gloves each of the following types:

- WL (=white and for left hand)
- WR (=white and for right hand)
- SL (=black and for left hand)
- SR (=black and for right hand)

In a draw of $k$ gloves, i.e., after $k$ "draws," success occurs if and only if two gloves of types WL and WR or two gloves of types SL and SR are among them. However, such pairings must occur no later than after $11(=5+5+1)$ draws, which already answers the question in a).

For b), we consider the probability of a failure after $k$ draws, where the smallest $k$ needs to be calculated for which this is 5, which is also confirmed by the calculation retrospectively.

For a failure to occur after $k \in\{6,7,8,9,10\}$ draws, only one of the four type combinations WL-SL, WL-SR, WR-SL, WR-SR may appear among the drawn gloves. The probability $P_{k}$ of a failure after $k$ draws is therefore

$$
P_{k}=4 \prod_{j=1}^{k} \frac{11-j}{21-j}
$$

and specifically for $k=6$ and $k=7$, one obtains the values

$$
P_{6} \approx 0.02167 \quad \text { or } \quad P_{7} \approx 0.00619
$$

The answer to the question in b) is thus $k=7$.