What is the expected value to gain in following game and how much one is expected to win if one plays $100$ $100$ games?

Question

What is the expected value to gain in following game and how much one is expected to win if one plays $100$ $100$ games?

In a card game return on getting an ace, other than club, is $$ 5$ $$5$ but if it is a club he gets extra $$ 10$ $$10$ and getting a club, other than ace, gets you $$ 1$ $$1$ .

1 Answer

Impact of this question

2633 views around the world

You can reuse this answer
Creative Commons License

Answer 1 · 2017-01-10T05:18:53

He is expected to win $$ 0.81$ $$0.81$ per game and

in $100$ $100$ games, he is expected to win $$ 80.77$ $$80.77$

Explanation:

When we have to find expected win or loss, what we have to do is identify probability of the event $p$ $p$ and multiply it with expected return $r$ $r$ .

If more than one events are there, which are mutually exclusive and their probabilities are $p_{1}, p_{2}, p_{3}, - - -$ $p_1,p_2,p_3,---$ and corresponding returns are $r_{1}, r_{2}, r_{3}, - - -$ $r_1,r_2,r_3,---$ and aggregate expectation is $\sum (p_{i} r_{i})$ $sum(p_ir_i)$ .

and if their are $n$ $n$ trials wins are $n \times \sum (p_{i} r_{i})$ $nxxsum(p_ir_i)$

Coming to the problem,

probability of getting an ace, other than club, is $\frac{3}{52}$ $3/52$ and return on it is $$ 5$ $$5$

probability of getting a club, other than ace, is $\frac{12}{52}$ $12/52$ and return on it is $$ 1$ $$1$

probability of getting ace of club is $\frac{15}{52}$ $15/52$ and return on it is $$ 15$ $$15$ , (here as he is expected to win $$ 5$ $$5$ for an ace and extra $$ 10$ $$10$ for club, he wins $$ 15$ $$15$ . If it is, however, $$ 1$ $$1$ for a club and extra $$ 10$ $$10$ for an ace, he wins $$ 11$ $$11$ ).

So for one game (considering $$ 15$ $$15$ for ace of club), one wins

$(\frac{3}{52} \times 5 + \frac{12}{52} \times 1 + \frac{1}{52} \times 15)$ $(3/52xx5+12/52xx1+1/52xx15)$

= $\frac{15 + 12 + 15}{52} = $ \frac{42}{52} = $ 0.81$ $(15+12+15)/52=$42/52=$0.81$

and in $100$ $100$ games, it is $100 \times \frac{42}{52} = $ 80.77$ $100xx42/52=$80.77$

Considering $$ 11$ $$11$ for ace of club, one wins

$(\frac{3}{52} \times 5 + \frac{12}{52} \times 1 + \frac{1}{52} \times 11)$ $(3/52xx5+12/52xx1+1/52xx11)$

= $\frac{15 + 12 + 11}{52} = $ \frac{38}{52} = $ 0.73$ $(15+12+11)/52=$38/52=$0.73$

and in $100$ $100$ games, it is $100 \times \frac{38}{52} = $ 73.08$ $100xx38/52=$73.08$

Science

Math

Humanities

... and beyond

What is the expected value to gain in following game and how much one is expected to win if one plays $100$ $100$ games?

In a card game return on getting an ace, other than club, is $$ 5$ $$5$ but if it is a club he gets extra $$ 10$ $$10$ and getting a club, other than ace, gets you $$ 1$ $$1$ .

1 Answer

Explanation:

Related questions

Impact of this question

Science

Math

Humanities

... and beyond

What is the expected value to gain in following game and how much one is expected to win if one plays 100100100 games?

In a card game return on getting an ace, other than club, is $5$5$5 but if it is a club he gets extra $10$10$10 and getting a club, other than ace, gets you $1$1$1.

1 Answer

Explanation:

Related questions

Impact of this question

What is the expected value to gain in following game and how much one is expected to win if one plays $100$ $100$ games?

In a card game return on getting an ace, other than club, is $$ 5$ $$5$ but if it is a club he gets extra $$ 10$ $$10$ and getting a club, other than ace, gets you $$ 1$ $$1$ .