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 2 times and lost 8 times.

1. Goat
2. Goat
3. Goat
4. Car
5. Goat
6. Goat
7. Goat
8. Goat
9. Goat
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 335 times and lost 665 times.

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