What is the expected number of boxes of cereals that he should buy?
$begingroup$
A company puts five different types of prizes into their cereal boxes,
one in each box and in equal proportions. If a customer decides to
collect all five prizes, what is the expected number of boxes of
cereals that he or she should buy?
TRY
Let $X$ be the number of boxes customer buys. For $i=1,2,3,4,5$, write
$$ X_{ij} = begin{cases} 1, & text{ith prize is inside jth box} \ 0, & text{otherwise} end{cases} $$
As I understand the problem, the number of boxes is not given so we may write
$$ X = sum_{j=1}^{infty} sum_{i=1}^5 X_{ij} $$
So
$$ E(X) = sum_{j geq 1 } sum_{i=1}^5 E(X_{ij}) $$
We know $E(X_{ij}) = P(X_{ij}=1)$ so we need to find proobability that ith prize is inside jth box. Here is the part where I get stuck. Am I appraoching this problem the correct way?
probability
asked Dec 15 '18 at 6:44
Jimmy SabaterJimmy Sabater
2,887324
$endgroup$
add a comment |
$begingroup$
A company puts five different types of prizes into their cereal boxes,
one in each box and in equal proportions. If a customer decides to
collect all five prizes, what is the expected number of boxes of
cereals that he or she should buy?
TRY
Let $X$ be the number of boxes customer buys. For $i=1,2,3,4,5$, write
$$ X_{ij} = begin{cases} 1, & text{ith prize is inside jth box} \ 0, & text{otherwise} end{cases} $$
As I understand the problem, the number of boxes is not given so we may write
$$ X = sum_{j=1}^{infty} sum_{i=1}^5 X_{ij} $$
So
$$ E(X) = sum_{j geq 1 } sum_{i=1}^5 E(X_{ij}) $$
We know $E(X_{ij}) = P(X_{ij}=1)$ so we need to find proobability that ith prize is inside jth box. Here is the part where I get stuck. Am I appraoching this problem the correct way?
probability
asked Dec 15 '18 at 6:44
Jimmy SabaterJimmy Sabater
2,887324
$endgroup$
add a comment |
$begingroup$
A company puts five different types of prizes into their cereal boxes,
one in each box and in equal proportions. If a customer decides to
collect all five prizes, what is the expected number of boxes of
cereals that he or she should buy?
TRY
Let $X$ be the number of boxes customer buys. For $i=1,2,3,4,5$, write
$$ X_{ij} = begin{cases} 1, & text{ith prize is inside jth box} \ 0, & text{otherwise} end{cases} $$
As I understand the problem, the number of boxes is not given so we may write
$$ X = sum_{j=1}^{infty} sum_{i=1}^5 X_{ij} $$
So
$$ E(X) = sum_{j geq 1 } sum_{i=1}^5 E(X_{ij}) $$
We know $E(X_{ij}) = P(X_{ij}=1)$ so we need to find proobability that ith prize is inside jth box. Here is the part where I get stuck. Am I appraoching this problem the correct way?
probability
asked Dec 15 '18 at 6:44
Jimmy SabaterJimmy Sabater
2,887324
$endgroup$
A company puts five different types of prizes into their cereal boxes,
one in each box and in equal proportions. If a customer decides to
collect all five prizes, what is the expected number of boxes of
cereals that he or she should buy?
TRY
Let $X$ be the number of boxes customer buys. For $i=1,2,3,4,5$, write
$$ X_{ij} = begin{cases} 1, & text{ith prize is inside jth box} \ 0, & text{otherwise} end{cases} $$
As I understand the problem, the number of boxes is not given so we may write
$$ X = sum_{j=1}^{infty} sum_{i=1}^5 X_{ij} $$
So
$$ E(X) = sum_{j geq 1 } sum_{i=1}^5 E(X_{ij}) $$
We know $E(X_{ij}) = P(X_{ij}=1)$ so we need to find proobability that ith prize is inside jth box. Here is the part where I get stuck. Am I appraoching this problem the correct way?
probability
probability
asked Dec 15 '18 at 6:44
Jimmy SabaterJimmy Sabater
2,887324
asked Dec 15 '18 at 6:44
Jimmy SabaterJimmy Sabater
2,887324
asked Dec 15 '18 at 6:44
Jimmy SabaterJimmy Sabater
2,887324
asked Dec 15 '18 at 6:44
Jimmy SabaterJimmy Sabater
2,887324
asked Dec 15 '18 at 6:44
Jimmy SabaterJimmy Sabater
2,887324
2,887324
add a comment |
add a comment |
1 Answer
1
active
oldest
votes
$begingroup$
HINT: Consider instead: if I have found a given number of unique prizes, what is the probability that the next box I open has a prize I don't have yet? What is the expected time to get a new prize?
SPOILER: this is the Coupon Collector's Problem.
answered Dec 15 '18 at 6:47
Dan UznanskiDan Uznanski
6,90021528
$endgroup$
add a comment |
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1 Answer
1
active
oldest
votes
1 Answer
1
active
oldest
votes
active
oldest
votes
active
oldest
votes
$begingroup$
HINT: Consider instead: if I have found a given number of unique prizes, what is the probability that the next box I open has a prize I don't have yet? What is the expected time to get a new prize?
SPOILER: this is the Coupon Collector's Problem.
answered Dec 15 '18 at 6:47
Dan UznanskiDan Uznanski
6,90021528
$endgroup$
add a comment |
$begingroup$
HINT: Consider instead: if I have found a given number of unique prizes, what is the probability that the next box I open has a prize I don't have yet? What is the expected time to get a new prize?
SPOILER: this is the Coupon Collector's Problem.
answered Dec 15 '18 at 6:47
Dan UznanskiDan Uznanski
6,90021528
$endgroup$
add a comment |
$begingroup$
HINT: Consider instead: if I have found a given number of unique prizes, what is the probability that the next box I open has a prize I don't have yet? What is the expected time to get a new prize?
SPOILER: this is the Coupon Collector's Problem.
answered Dec 15 '18 at 6:47
Dan UznanskiDan Uznanski
6,90021528
$endgroup$
HINT: Consider instead: if I have found a given number of unique prizes, what is the probability that the next box I open has a prize I don't have yet? What is the expected time to get a new prize?
SPOILER: this is the Coupon Collector's Problem.
answered Dec 15 '18 at 6:47
Dan UznanskiDan Uznanski
6,90021528
answered Dec 15 '18 at 6:47
Dan UznanskiDan Uznanski
6,90021528
answered Dec 15 '18 at 6:47
Dan UznanskiDan Uznanski
6,90021528
answered Dec 15 '18 at 6:47
Dan UznanskiDan Uznanski
6,90021528
6,90021528
add a comment |
add a comment |
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