Two different coins are flipped a different number of times. It is known that one of the coins is biased.

\nThe results for both coins are given in the table below.

\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\nOutcome | Coin 1 | Coin 2 |
---|---|---|

Heads | $\\var{h1}$ | $\\var{h2}$ |

Tails | $\\var{t1}$ | $\\var{t2}$ |

Money won in part b.

", "definition": "random(5..20 #5)", "name": "win", "templateType": "anything"}, "t2": {"group": "Ungrouped variables", "description": "Number of tails for coin 2.

", "definition": "random(2,3)", "name": "t2", "templateType": "anything"}, "h1": {"group": "Ungrouped variables", "description": "number of heads for coin 1.

", "definition": "random(6000..7000 #250)", "name": "h1", "templateType": "anything"}, "h2": {"group": "Ungrouped variables", "description": "Number of heads for coin 2.

", "definition": "15 - t2", "name": "h2", "templateType": "anything"}, "t1": {"group": "Ungrouped variables", "description": "Number of tails for coin 1

", "definition": "10000 - h1", "name": "t1", "templateType": "anything"}}, "functions": {}, "preamble": {"css": "", "js": ""}, "metadata": {"licence": "Creative Commons Attribution 4.0 International", "description": "This question aims to assess the student's understanding of the difference between biased and unbiased events and also to assess the student's understanding of the fact that the experimental probability tends towards the theoretical probability as the number of trials increases.

"}, "parts": [{"variableReplacementStrategy": "originalfirst", "marks": 0, "type": "gapfill", "gaps": [{"variableReplacementStrategy": "originalfirst", "scripts": {}, "marks": 1, "correctAnswerStyle": "plain", "showCorrectAnswer": true, "correctAnswerFraction": true, "mustBeReduced": false, "mustBeReducedPC": 0, "notationStyles": ["plain", "en", "si-en"], "type": "numberentry", "maxValue": "{h1}/10000", "showFeedbackIcon": true, "minValue": "{h1}/10000", "variableReplacements": [], "allowFractions": true}, {"variableReplacementStrategy": "originalfirst", "scripts": {}, "marks": 1, "correctAnswerStyle": "plain", "showCorrectAnswer": true, "correctAnswerFraction": true, "mustBeReduced": false, "mustBeReducedPC": 0, "notationStyles": ["plain", "en", "si-en"], "type": "numberentry", "maxValue": "{h2}/15", "showFeedbackIcon": true, "minValue": "{h2}/15", "variableReplacements": [], "allowFractions": true}], "prompt": "Calculate the experimental probability of tossing heads, $P(\\text{heads})$, for each coin. Enter your answers as fractions.

\n$P(\\text{heads})\\; \\text{for Coin $1$} =$ [[0]]

\n$P(\\text{heads})\\; \\text{for Coin $2$} =$ [[1]]

\n", "showFeedbackIcon": true, "showCorrectAnswer": true, "variableReplacements": [], "scripts": {}}, {"choices": ["Coin 1

", "Coin 2

"], "distractors": ["", ""], "marks": 0, "stepsPenalty": 0, "shuffleChoices": false, "steps": [{"variableReplacementStrategy": "originalfirst", "marks": 0, "type": "information", "prompt": "We say that an event is biased if one outcome of the event is more likely than the other outcomes.

", "showFeedbackIcon": true, "showCorrectAnswer": true, "variableReplacements": [], "scripts": {}}], "prompt": "Based on the available results and the experimental probabilities calculated in part **a)**, for which coin is there more evidence of bias?

Coin 1

", "Coin 2

"], "distractors": ["", ""], "marks": 0, "shuffleChoices": false, "variableReplacementStrategy": "originalfirst", "prompt": "You decide to take part in a bet with your friend Alex.

\nAlex says that if you toss either one of the above coins and the coin lands on tails then you win $£\\var{win}$. However, if the coin lands on heads then Alex wins $£\\var{win}$.

\nWhich coin should you choose in order to have a more reliable chance of winning the bet?

", "displayColumns": 0, "showCorrectAnswer": true, "maxMarks": 0, "displayType": "radiogroup", "variableReplacements": [], "type": "1_n_2", "matrix": [0, "1"], "showFeedbackIcon": true, "minMarks": 0, "scripts": {}}], "type": "question", "advice": "The results from the tosses of both coins are given in the table below.

\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\nOutcome | Coin 1 | Coin 2 |

Heads | $\\var{h1}$ | $\\var{h2}$ |

Tails | $\\var{t1}$ | $\\var{t2}$ |

We can calculate the number of tosses of each coin by adding together the number of heads and tails obtained for each coin.

\nCoin $1$ was tossed $\\var{h1+t1}$ times.

\nCoin $2$ was tossed $\\var{h2+t2}$ times.

\nThis means that for Coin $1$, the experimental probability of tossing heads is

\n\\[

\\begin{align}

\\displaystyle\\frac{\\var{h1}}{10000} &= \\var{{h1}/10000}\\\\

&= \\var{100*({h1}/10000)}\\%.

\\end{align}

\\]

Whereas for Coin $2$, the experimental probability of tossing heads is

\n\\[\\begin{align}\\displaystyle\\frac{\\var{h2}}{15} &= \\var{dpformat({h2}/15,2)} \\; (\\text{rounded to two decimal places})\\\\&= \\var{dpformat(100*({h2}/15),0)}\\%. \\end{align} \\]

\n\\[\\begin{align}\\displaystyle\\frac{\\var{h2}}{15} &= \\var{{h2}/15}\\\\&= \\var{100*({h2}/15)}\\%. \\end{align} \\]

\nWe can see from part **a)** that the number of tosses of Coin $1$ is much larger than the number of tosses of Coin $2$.

It is important to know that as the number of trials in an experiment gets very large the experimental probability **tends towards** the theoretical probability.

For an unbiased coin, the theoretical probability of tossing heads is $50\\%$ so as the number of tosses gets very large we would expect the experimental probability of tossing heads to get closer to $50\\%$.

\nTherefore, as the number of tosses of Coin $1$ is very large and the experimental probability of tossing heads with this coin is significantly different from $50\\%$, then it is quite likely that Coin $1$ could be biased.

\nOn the other hand, as the number of tosses for Coin $2$ is small, we cannot give an accurate opinion of whether this coin is biased or unbiased.

\nSo, there is more evidence of bias for Coin $1$ than for Coin $2$.

\nAs discussed in part **b)**, there is more evidence of bias for Coin $1$ than for Coin $2$.

Furthermore, we saw that the experimental probability of tossing heads with Coin $1$ is significantly different from $50\\%$, which means that the coin could be biased in favour of heads.

\nSo, as we want the coin to land tails then we want to choose the coin that has the most reliable chance of landing tails-up.

\nSince there is less evidence of bias for Coin $2$ than for Coin $1$, we should choose to use Coin $2$ in the bet as there is a more reliable chance of tossing tails with this coin.

", "rulesets": {}, "extensions": [], "name": "Which coin is more likely to be biased?", "variablesTest": {"condition": "", "maxRuns": 100}, "contributors": [{"name": "Christian Lawson-Perfect", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/7/"}, {"name": "Elliott Fletcher", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/1591/"}, {"name": "Xiaodan Leng", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/2146/"}]}]}], "contributors": [{"name": "Christian Lawson-Perfect", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/7/"}, {"name": "Elliott Fletcher", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/1591/"}, {"name": "Xiaodan Leng", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/2146/"}]}