Goat or Car? Game Show Simulator

A player is presented with three doors. Behind one door is a new car. The other two doors conceal a goat. The player chooses a door then the host reveals a goat behind one of the two remaining doors. Should the player maintain his choice or switch to the only remaining door?

My intuition suggested that once the other goat was revealed that it became a simple 50% chance that the player had chosen correctly and, therefore, there would be no statistical benefit in changing their mind. In the simulation below, the player always sticks with their original choice.

Player won 6 times and lost 4 times.

1. Goat
2. Car
3. Car
4. Goat
5. Car
6. Car
7. Goat
8. Goat
9. Car
10. Car

Every refresh of this page runs the test again so you will see variation in results. Of course, 10 iterations are statistically insignificant. A simulation of 1,000 iterations makes the ratio clearer:

Player won 321 times and lost 679 times.

1. Car
2. Car
3. Goat
4. Goat
5. Car
6. Goat
7. Car
8. Goat
9. Goat
10. Goat
11. Goat
12. Goat
13. Goat
14. Car
15. Goat
16. Goat
17. Car
18. Goat
19. Goat
20. Car
21. Goat
22. Goat
23. Car
24. Goat
25. Car
26. Car
27. Goat
28. Goat
29. Car
30. Car
31. Car
32. Goat
33. Car
34. Car
35. Goat
36. Car
37. Goat
38. Car
39. Car
40. Car
41. Goat
42. Goat
43. Car
44. Goat
45. Car
46. Goat
47. Goat
48. Goat
49. Goat
50. Goat
51. Goat
52. Goat
53. Car
54. Goat
55. Goat
56. Goat
57. Goat
58. Goat
59. Goat
60. Goat
61. Car
62. Goat
63. Goat
64. Goat
65. Car
66. Goat
67. Goat
68. Car
69. Car
70. Goat
71. Goat
72. Goat
73. Goat
74. Goat
75. Car
76. Goat
77. Goat
78. Goat
79. Car
80. Goat
81. Car
82. Car
83. Goat
84. Goat
85. Goat
86. Goat
87. Goat
88. Car
89. Car
90. Goat
91. Car
92. Goat
93. Car
94. Goat
95. Goat
96. Goat
97. Car
98. Goat
99. Goat
100. Goat
101. Goat
102. Goat
103. Goat
104. Goat
105. Car
106. Goat
107. Goat
108. Goat
109. Goat
110. Car
111. Car
112. Car
113. Car
114. Goat
115. Goat
116. Goat
117. Goat
118. Car
119. Goat
120. Goat
121. Goat
122. Goat
123. Goat
124. Goat
125. Car
126. Goat
127. Goat
128. Goat
129. Goat
130. Goat
131. Goat
132. Car
133. Car
134. Goat
135. Goat
136. Goat
137. Car
138. Goat
139. Car
140. Goat
141. Car
142. Goat
143. Goat
144. Car
145. Goat
146. Goat
147. Goat
148. Goat
149. Goat
150. Goat
151. Goat
152. Goat
153. Car
154. Goat
155. Goat
156. Car
157. Car
158. Goat
159. Goat
160. Goat
161. Goat
162. Goat
163. Car
164. Car
165. Car
166. Car
167. Goat
168. Goat
169. Car
170. Goat
171. Car
172. Goat
173. Goat
174. Goat
175. Car
176. Goat
177. Car
178. Goat
179. Goat
180. Car
181. Car
182. Goat
183. Goat
184. Goat
185. Goat
186. Goat
187. Car
188. Goat
189. Goat
190. Car
191. Goat
192. Goat
193. Car
194. Car
195. Goat
196. Goat
197. Goat
198. Goat
199. Goat
200. Car
201. Car
202. Goat
203. Goat
204. Goat
205. Car
206. Car
207. Goat
208. Goat
209. Goat
210. Goat
211. Goat
212. Goat
213. Goat
214. Goat
215. Car
216. Goat
217. Goat
218. Goat
219. Car
220. Goat
221. Car
222. Goat
223. Goat
224. Car
225. Goat
226. Goat
227. Car
228. Goat
229. Goat
230. Goat
231. Car
232. Car
233. Goat
234. Car
235. Car
236. Goat
237. Car
238. Goat
239. Car
240. Goat
241. Car
242. Goat
243. Goat
244. Goat
245. Goat
246. Goat
247. Goat
248. Goat
249. Car
250. Goat
251. Car
252. Goat
253. Goat
254. Goat
255. Goat
256. Car
257. Goat
258. Goat
259. Car
260. Goat
261. Car
262. Goat
263. Goat
264. Car
265. Goat
266. Goat
267. Car
268. Goat
269. Goat
270. Car
271. Goat
272. Car
273. Goat
274. Goat
275. Goat
276. Car
277. Goat
278. Car
279. Car
280. Car
281. Goat
282. Car
283. Car
284. Goat
285. Goat
286. Car
287. Goat
288. Goat
289. Car
290. Goat
291. Car
292. Goat
293. Goat
294. Car
295. Car
296. Goat
297. Car
298. Car
299. Goat
300. Car
301. Car
302. Goat
303. Goat
304. Goat
305. Goat
306. Car
307. Goat
308. Goat
309. Goat
310. Goat
311. Goat
312. Goat
313. Car
314. Goat
315. Car
316. Goat
317. Goat
318. Goat
319. Car
320. Goat
321. Car
322. Car
323. Goat
324. Car
325. Car
326. Goat
327. Car
328. Goat
329. Goat
330. Goat
331. Goat
332. Goat
333. Goat
334. Car
335. Goat
336. Car
337. Goat
338. Goat
339. Goat
340. Goat
341. Goat
342. Goat
343. Goat
344. Goat
345. Goat
346. Car
347. Goat
348. Goat
349. Goat
350. Car
351. Car
352. Goat
353. Goat
354. Goat
355. Car
356. Goat
357. Goat
358. Goat
359. Car
360. Goat
361. Goat
362. Goat
363. Car
364. Goat
365. Car
366. Goat
367. Car
368. Goat
369. Goat
370. Goat
371. Goat
372. Goat
373. Car
374. Goat
375. Car
376. Car
377. Car
378. Goat
379. Car
380. Goat
381. Goat
382. Goat
383. Goat
384. Goat
385. Car
386. Goat
387. Goat
388. Goat
389. Goat
390. Car
391. Goat
392. Goat
393. Goat
394. Car
395. Car
396. Goat
397. Car
398. Goat
399. Goat
400. Car
401. Goat
402. Car
403. Car
404. Car
405. Car
406. Goat
407. Goat
408. Goat
409. Car
410. Car
411. Car
412. Goat
413. Car
414. Car
415. Goat
416. Goat
417. Goat
418. Car
419. Car
420. Goat
421. Goat
422. Car
423. Goat
424. Goat
425. Car
426. Goat
427. Car
428. Goat
429. Car
430. Goat
431. Goat
432. Goat
433. Car
434. Goat
435. Car
436. Goat
437. Goat
438. Car
439. Goat
440. Goat
441. Goat
442. Goat
443. Goat
444. Car
445. Car
446. Goat
447. Goat
448. Car
449. Goat
450. Goat
451. Goat
452. Goat
453. Goat
454. Goat
455. Goat
456. Goat
457. Goat
458. Goat
459. Goat
460. Goat
461. Goat
462. Car
463. Goat
464. Car
465. Goat
466. Goat
467. Goat
468. Car
469. Car
470. Goat
471. Car
472. Car
473. Goat
474. Goat
475. Goat
476. Goat
477. Car
478. Goat
479. Goat
480. Goat
481. Goat
482. Goat
483. Goat
484. Goat
485. Goat
486. Car
487. Car
488. Goat
489. Goat
490. Car
491. Car
492. Goat
493. Goat
494. Goat
495. Goat
496. Goat
497. Car
498. Car
499. Goat
500. Goat
501. Car
502. Goat
503. Goat
504. Goat
505. Goat
506. Car
507. Goat
508. Goat
509. Goat
510. Goat
511. Goat
512. Car
513. Goat
514. Car
515. Car
516. Car
517. Goat
518. Goat
519. Goat
520. Goat
521. Goat
522. Goat
523. Goat
524. Goat
525. Goat
526. Car
527. Goat
528. Car
529. Goat
530. Car
531. Goat
532. Car
533. Goat
534. Goat
535. Goat
536. Car
537. Goat
538. Goat
539. Car
540. Goat
541. Goat
542. Goat
543. Car
544. Goat
545. Goat
546. Goat
547. Car
548. Car
549. Goat
550. Goat
551. Goat
552. Car
553. Goat
554. Car
555. Car
556. Car
557. Car
558. Goat
559. Car
560. Car
561. Car
562. Car
563. Goat
564. Car
565. Car
566. Car
567. Goat
568. Goat
569. Car
570. Car
571. Goat
572. Goat
573. Goat
574. Goat
575. Car
576. Goat
577. Car
578. Car
579. Goat
580. Goat
581. Goat
582. Goat
583. Goat
584. Goat
585. Goat
586. Car
587. Goat
588. Goat
589. Goat
590. Goat
591. Car
592. Goat
593. Goat
594. Goat
595. Car
596. Goat
597. Goat
598. Goat
599. Goat
600. Goat
601. Goat
602. Goat
603. Car
604. Goat
605. Goat
606. Car
607. Goat
608. Goat
609. Goat
610. Goat
611. Car
612. Goat
613. Car
614. Goat
615. Car
616. Goat
617. Car
618. Goat
619. Goat
620. Car
621. Goat
622. Goat
623. Goat
624. Goat
625. Goat
626. Car
627. Goat
628. Car
629. Goat
630. Goat
631. Goat
632. Goat
633. Goat
634. Car
635. Goat
636. Goat
637. Goat
638. Goat
639. Goat
640. Car
641. Goat
642. Goat
643. Car
644. Car
645. Goat
646. Car
647. Goat
648. Goat
649. Car
650. Goat
651. Car
652. Car
653. Goat
654. Goat
655. Goat
656. Goat
657. Car
658. Goat
659. Goat
660. Goat
661. Goat
662. Goat
663. Goat
664. Car
665. Goat
666. Goat
667. Goat
668. Goat
669. Car
670. Car
671. Car
672. Goat
673. Goat
674. Goat
675. Goat
676. Goat
677. Goat
678. Goat
679. Car
680. Goat
681. Goat
682. Goat
683. Goat
684. Car
685. Goat
686. Goat
687. Goat
688. Goat
689. Car
690. Goat
691. Goat
692. Goat
693. Goat
694. Car
695. Goat
696. Goat
697. Goat
698. Goat
699. Goat
700. Goat
701. Goat
702. Car
703. Car
704. Goat
705. Car
706. Goat
707. Goat
708. Goat
709. Goat
710. Goat
711. Goat
712. Goat
713. Car
714. Car
715. Goat
716. Goat
717. Car
718. Goat
719. Goat
720. Goat
721. Goat
722. Goat
723. Goat
724. Goat
725. Goat
726. Goat
727. Car
728. Goat
729. Goat
730. Goat
731. Goat
732. Goat
733. Goat
734. Goat
735. Goat
736. Car
737. Goat
738. Goat
739. Goat
740. Goat
741. Goat
742. Car
743. Goat
744. Goat
745. Goat
746. Car
747. Goat
748. Goat
749. Car
750. Goat
751. Goat
752. Goat
753. Goat
754. Goat
755. Goat
756. Car
757. Car
758. Car
759. Car
760. Goat
761. Goat
762. Car
763. Goat
764. Car
765. Goat
766. Car
767. Goat
768. Goat
769. Car
770. Goat
771. Car
772. Car
773. Car
774. Goat
775. Goat
776. Car
777. Goat
778. Goat
779. Goat
780. Goat
781. Car
782. Car
783. Goat
784. Goat
785. Goat
786. Goat
787. Goat
788. Car
789. Car
790. Goat
791. Goat
792. Goat
793. Goat
794. Goat
795. Goat
796. Goat
797. Goat
798. Goat
799. Car
800. Goat
801. Goat
802. Car
803. Car
804. Goat
805. Goat
806. Goat
807. Goat
808. Goat
809. Goat
810. Car
811. Car
812. Goat
813. Car
814. Goat
815. Goat
816. Goat
817. Goat
818. Car
819. Car
820. Goat
821. Goat
822. Car
823. Goat
824. Goat
825. Car
826. Goat
827. Goat
828. Car
829. Goat
830. Goat
831. Goat
832. Goat
833. Goat
834. Goat
835. Goat
836. Goat
837. Car
838. Goat
839. Car
840. Goat
841. Car
842. Goat
843. Goat
844. Goat
845. Car
846. Car
847. Car
848. Goat
849. Goat
850. Goat
851. Goat
852. Car
853. Goat
854. Car
855. Goat
856. Car
857. Goat
858. Goat
859. Goat
860. Car
861. Goat
862. Car
863. Goat
864. Goat
865. Goat
866. Goat
867. Goat
868. Goat
869. Goat
870. Goat
871. Goat
872. Goat
873. Car
874. Car
875. Goat
876. Goat
877. Goat
878. Car
879. Goat
880. Goat
881. Goat
882. Goat
883. Car
884. Goat
885. Goat
886. Goat
887. Goat
888. Car
889. Car
890. Car
891. Car
892. Goat
893. Goat
894. Goat
895. Goat
896. Goat
897. Car
898. Car
899. Car
900. Goat
901. Car
902. Goat
903. Car
904. Goat
905. Car
906. Goat
907. Goat
908. Goat
909. Goat
910. Goat
911. Goat
912. Car
913. Goat
914. Goat
915. Car
916. Car
917. Goat
918. Goat
919. Goat
920. Goat
921. Goat
922. Goat
923. Car
924. Goat
925. Goat
926. Goat
927. Goat
928. Car
929. Goat
930. Goat
931. Goat
932. Goat
933. Goat
934. Car
935. Car
936. Goat
937. Car
938. Goat
939. Car
940. Goat
941. Goat
942. Car
943. Goat
944. Car
945. Goat
946. Car
947. Goat
948. Goat
949. Goat
950. Car
951. Car
952. Car
953. Goat
954. Goat
955. Goat
956. Goat
957. Goat
958. Goat
959. Car
960. Goat
961. Goat
962. Goat
963. Car
964. Car
965. Goat
966. Car
967. Goat
968. Goat
969. Goat
970. Car
971. Goat
972. Goat
973. Goat
974. Goat
975. Goat
976. Goat
977. Goat
978. Car
979. Goat
980. Goat
981. Goat
982. Goat
983. Goat
984. Goat
985. Goat
986. Goat
987. Car
988. Goat
989. Car
990. Car
991. Goat
992. Goat
993. Goat
994. Goat
995. Car
996. Car
997. Car
998. Goat
999. Goat
1000. Car

But what if there were 10 doors with nine goats and one car? The game would still work the same where the player picks one door, and the host reveals 8 goats. Should the player switch? Here is a 1,000 iteration simulation where there are 10 doors and the player always sticks with their original choice:

Player won 84 times and lost 916 times.

So clearly, the number of doors influences the probability that the player would be better of by switching to the other remaining door.

Would you like to run a simulation? Enter your variables below:

Custom test result

Player won 31 times and lost 69 times.

Methodology

This page uses the following PHP function to run the simulations:

function runTest($iterations, $number_of_doors=3, $player_stays=true, $show_pictures=false){

	// sanitize iterations
	$iterations = (int) preg_replace("/[^0-9]/","",$iterations);
	$iterations = ($iterations < 1)? 10 : $iterations;
	$iterations = ($iterations > 10000)? 10000 : $iterations;

	// sanitize number_of_doors
	$number_of_doors = (int) preg_replace("/[^0-9]/","",$number_of_doors);
	$number_of_doors = ($number_of_doors < 3)? 3 : $number_of_doors;
	$number_of_doors = ($number_of_doors > 10000)? 10000 : $number_of_doors;

	// generate a standard array
	$doors_original[0] = 'car';
	for($i=1; $i < $number_of_doors ; $i++){
		$doors_original[$i] = 'goat';
	}
	
	// initialize our scores
	$losses = 0; $wins = 0;
	
	// decide how big the images will be
	if($iterations >= 1000){
		$size = '24';
	} else if($iterations >= 100){
		$size = '32';
	} else {
		$size = '64';
	}
	
	$output = '<ol>';
	for($i=0; $i<$iterations; $i++){
		// make a temporary copy for this usage
		$doors = $doors_original;
		
		// shuffle the array
		shuffle($doors);
		
		// player always picks door number 1
		$players_first_choice = $doors[0];
		
		// did the player win?
		if($players_first_choice == 'car' && $player_stays==true){
			$output .= '<li class="c'.$size.'"><b>Car</b></li>';
			$wins++;
		} else if($players_first_choice == 'goat' && $player_stays==true){
			$output .= '<li class="g'.$size.'"><i>Goat</i></li>';
			$losses++;
		} else if($players_first_choice == 'car' && $player_stays==false){
			$output .= '<li class="g'.$size.'"><i>Goat</i></li>';
			$losses++;
		} else if($players_first_choice == 'goat' && $player_stays==false){
			$output .= '<li class="c'.$size.'"><b>Car</b></li>';
			$wins++;
		}
	}
	$output .= '</ol><div class="clear"> </div>';
	
	echo '<h3>Player won '.$wins.' time'.(($wins >1)? 's':'').
	     ' and lost '.$losses.' time'.(($losses >1)? 's':'').'.</h3>';
	if($show_pictures){
		echo $output;
	}
}