// Numbas version: finer_feedback_settings {"name": "Michael's copy of Probability -- choosing digits with replacement - highest digit", "extensions": [], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "question_groups": [{"pickingStrategy": "all-ordered", "questions": [{"name": "Michael's copy of Probability -- choosing digits with replacement - highest digit", "advice": "\n

a) There are $\\var{d+1}$ different digits and so there are $\\var{d+1}^{\\var{r}}=\\var{(d+1)^r}$ different seqences of length $\\var{r}$.

However, if we consider sequences in which the digits do not exceed $\\var{k}$ ,

then there are $\\var{k+1}$ different possible digits in the sequence, and so there are $\\var{k+1}^{\\var{r}}=\\var{(k+1)^r}$ possible sequences.

So \\[\\begin{eqnarray*} \\mbox{Probability} &=& \\frac{\\mbox{number of successes}}{\\mbox{number of trials}}\\\\ &=& \\frac{\\var{(k+1)^r}}{\\var{(d+1)^r}}\\\\ &=& \\var{ans1} \\end{eqnarray*} \\] to $3$ decimal places.

b) Number $N$ of sequences with largest digit $\\var{k}$ is given by:

\\[\\begin{eqnarray*} N &=&\\mbox{number with largest digit}\\;\\le \\var{k} - \\mbox{number with largest digit}\\;\\le\\var{k-1} \\\\ &=& \\var{k+1}^{\\var{r}}-\\var{k}^{\\var{r}}\\\\ &=& \\var{(k+1)^r}-\\var{k^r}\\\\ &=& \\var{(k+1)^r-k^r}\\\\ \\mbox{So probability} &=& \\frac{\\mbox{number of successes}}{\\mbox{number of trials}} =\\frac{N}{\\mbox{number of trials}}\\\\ &=& \\frac{\\var{(k+1)^r-k^r}}{\\var{(d+1)^r}}\\\\ &=& \\var{ans2} \\end{eqnarray*} \\] to $3$ decimal places.

\n ", "tags": [], "statement": "\n

Suppose a sequence of $\\var{r}$ digits is picked at random (with replacement) from the set
\\[\\{0,\\;1,\\;2,\\ldots,\\;\\var{d}\\}\\]

What is the probability that:

\n ", "functions": {}, "extensions": [], "ungrouped_variables": ["d", "ans1", "k", "r", "t", "tol", "tans1", "ans2", "tans2", "test"], "preamble": {"js": "", "css": ""}, "rulesets": {"std": ["all", "fractionNumbers", "!collectNumbers", "!noLeadingMinus"]}, "parts": [{"marks": 0, "customMarkingAlgorithm": "", "variableReplacements": [], "prompt": "\n

No digit in the sequence exceeds $\\var{k}$?

Probability =[[0]]? (to $3$ decimal places)

The largest digit in the sequence is $\\var{k}$?

Probability =[[0]]? (to $3$ decimal places)

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r digits are picked at random (with replacement) from the set $\\{0,\\;1,\\;2,\\ldots,\\;n\\}$. Probabilities that 1) all $\\lt k$, 2) largest is $k$?

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