Video: AH5P1-Q14-189198704307

In a game, a fair spinner has four sections. a) Marcia writes, “Since there are four colors, the probability of landing on yellow must be 1/4, so the probability of getting two yellows is 1/4 + 1/4 = 1/2.” Make two criticisms of Marcia’s method.

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Video Transcript

In a game, a fair spinner has four sections. Part a) Marcia writes, “Since there are four colors, the probability of landing on yellow must be one-quarter, so the probability of getting two yellows is one-quarter plus one-quarter is equal to one-half.” Make two criticisms of Marcia’s method.

There is also a part b) to this question which we will look at later. If all four sections were the same size, in this case 90 degrees, then yes, Marcia would be correct. The probability of landing on yellow would be one-quarter. However, we can see from the diagram that the sections are different sizes. For example, the yellow section is 120 degrees. This means that the probability of landing on yellow will be 120 out of 360 as there are 360 degrees in a circle.

This fraction can be simplified by dividing the numerator and denominator by 10. Remember whatever you do to the top, you must do to the bottom. This gives us 12 over 36. We can divide the numerator and denominator of this new fraction by 12. 12 divided by 12 is equal to one and 36 divided by 12 is equal to three. This means that the probability of landing on the yellow section is one-third or one out of three.

This means that one criticism of Marcia’s method would be the chance of landing on each color is not the same. For example, the probability of landing on a yellow is one-third, not one-quarter. This means that the first part of Marcia’s sentence is incorrect.

Marcia also stated that the probability of getting two yellows was a quarter plus a quarter which is equal to a half. However, we know that the AND rule in probability means multiply. Therefore, the probability of getting a yellow and a second yellow would involve multiplying the two probabilities.

We can therefore say that a second criticism of Marcia’s method is as follows: the probability of getting two yellows is the probability of getting a yellow multiplied by the probability of getting yellow, not adding the probabilities. Marcia’s mistake is that she added instead of multiplying.

The second part of the question says the following. b) The probability of getting two greens from two spins is one out of 36. Work out the angle of the green sector.

In order to answer this question, we need to work out the angle of the green sector labelled 𝜃. Before we can do this, we need to work out the probability of landing on the green section. If we let the probability of landing on the green section equal 𝑥, then we know that the probability of getting two greens — green and green — is 𝑥 multiplied by 𝑥. This is because the word “and” means multiply in probability as mentioned in part a). 𝑥 multiplied by 𝑥 is equal to 𝑥 squared. Therefore, the probability of getting two greens is equal to 𝑥 squared.

We were told in the question that the probability of getting two greens from two spins is one out of 36. Therefore, 𝑥 squared is equal to one out of 36. The opposite or inverse of squaring is square rooting. We can therefore square root both sides of this equation to work out 𝑥. 𝑥 is equal to the square root of one over 36. Normally, we would include a plus and a minus in front of the square root as square rooting a number can give two answers. However, as we’re dealing with probability, our answer must be positive. This means that 𝑥 must be the positive square root of one over 36 which we can leave as root one over 36.

Our laws of surds tell us that the square root of 𝑎 divided by 𝑏 is equal to the square root of 𝑎 divided by the square root of 𝑏. This means that 𝑥 is equal to the square root of one divided by the square root of 36. The square root of one is equal to one and the square root of 36 is equal to six as six multiplied by six equals 36. We can therefore say that the probability that the spinner lands on green is one out of six or one-sixth.

We’ve not quite finished the question however as we need to work out the angle of the green sector. The angle 𝜃 out of 360 must be equal to one-sixth. Multiplying both sides of this equation by 360 gives us 𝜃 is equal to one-sixth multiplied by 360. We can calculate one-sixth multiplied by 360 or one-sixth of 360 by dividing 360 by six. 36 divided by six is equal to six and zero divided by six is equal to zero. Therefore, 360 divided by six equals 60.

This means that the angle of the green sector is equal to 60 degrees. Whilst the diagram hasn’t been drawn to scale, this seems like a sensible answer as it is clear from the diagram that green is less than 90 degrees.

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